121-402: Keno / k iː n oʊ / is a lottery -like gambling game often played at modern casinos , and also offered as a game in some lotteries. Players wager by choosing numbers ranging from 1 through (usually) 80. After all players make their wagers, 20 numbers (some variants draw fewer numbers) are drawn at random, either with a ball machine similar to ones used for lotteries and bingo , or with
242-717: A double exponential function . Its growth rate is similar to n n {\displaystyle n^{n}} , but slower by an exponential factor. One way of approaching this result is by taking the natural logarithm of the factorial, which turns its product formula into a sum, and then estimating the sum by an integral: ln n ! = ∑ x = 1 n ln x ≈ ∫ 1 n ln x d x = n ln n − n + 1. {\displaystyle \ln n!=\sum _{x=1}^{n}\ln x\approx \int _{1}^{n}\ln x\,dx=n\ln n-n+1.} Exponentiating
363-521: A random number generator . Each casino sets its own series of payouts, called "paytables". The player is paid based on how many numbers were chosen (either player selection, or the terminal picking the numbers), the number of matches out of those chosen, and the wager. There are a wide variety of keno paytables depending on the casino, usually with a larger " house edge " than other games, ranging from less than 4 percent to over 35 percent in online play, and 20-40% in in-person casinos. By way of comparison,
484-408: A 1685 treatise by John Wallis , a study of their approximate values for large values of n {\displaystyle n} by Abraham de Moivre in 1721, a 1729 letter from James Stirling to de Moivre stating what became known as Stirling's approximation , and work at the same time by Daniel Bernoulli and Leonhard Euler formulating the continuous extension of the factorial function to
605-540: A 20 spot ticket is approximately 1 in 3.5 quintillion (1 in 3,535,316,142,212,174,320). Even though it is highly improbable to hit all 20 numbers on a 20 spot ticket, the same player would typically also get paid for hitting “catches” 0, 1, 2, 3, and 7 through 19 out of 20, often with the 17 through 19 catches paying the same as the solid 20 hit. Some of the other paying "catches" on a 20 spot ticket or any other ticket with high "solid catch" odds are in reality very possible to hit: Probabilities change significantly based on
726-467: A common example in the use of different computer programming styles and methods. The computation of n ! {\displaystyle n!} can be expressed in pseudocode using iteration as or using recursion based on its recurrence relation as Other methods suitable for its computation include memoization , dynamic programming , and functional programming . The computational complexity of these algorithms may be analyzed using
847-545: A definition for the factorial at all complex numbers other than the negative integers. One property of the gamma function, distinguishing it from other continuous interpolations of the factorials, is given by the Bohr–Mollerup theorem , which states that the gamma function (offset by one) is the only log-convex function on the positive real numbers that interpolates the factorials and obeys the same functional equation. A related uniqueness theorem of Helmut Wielandt states that
968-421: A factor of two to produce one of these trailing zeros. The leading digits of the factorials are distributed according to Benford's law . Every sequence of digits, in any base, is the sequence of initial digits of some factorial number in that base. Another result on divisibility of factorials, Wilson's theorem , states that ( n − 1 ) ! + 1 {\displaystyle (n-1)!+1}
1089-697: A function of the number of digits or bits in the result. By Stirling's formula, n ! {\displaystyle n!} has b = O ( n log n ) {\displaystyle b=O(n\log n)} bits. The Schönhage–Strassen algorithm can produce a b {\displaystyle b} -bit product in time O ( b log b log log b ) {\displaystyle O(b\log b\log \log b)} , and faster multiplication algorithms taking time O ( b log b ) {\displaystyle O(b\log b)} are known. However, computing
1210-519: A group of 51 society lotteries across the United Kingdom with a common drawing and prize pool. Each drawing is held on behalf of one or more of the society lotteries, whose revenues go to support health-related causes in their respective area. The Health Lottery received criticism on launch for only pledging to donate 20.3% of ticket costs to charity, compared to the National Lottery's 28%, and that
1331-547: A lottery offering tickets for sale is the lottery organized by Roman Emperor Augustus . The funds were for repairs in the City of Rome, and the winners were given prizes in the form of articles of unequal value. The first recorded lotteries to offer tickets for sale with prizes in the form of money were held in the Low Countries in the 15th century. Various towns held public lotteries to raise money for town fortifications, and to help
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#17328522999141452-634: A lottery to raise money to purchase cannons for the defense of Philadelphia. Several of these lotteries offered prizes in the form of " Pieces of Eight ". George Washington 's Mountain Road Lottery in 1768 was unsuccessful, but these rare lottery tickets bearing Washington's signature became collectors' items; one example sold for about $ 15,000 in 2007. Washington was also a manager for Col. Bernard Moore 's "Slave Lottery" in 1769, which advertised land and slaves as prizes in The Virginia Gazette . At
1573-487: A modified form of the factorial, omitting the factors in the factorial that are divisible by p . The digamma function is the logarithmic derivative of the gamma function. Just as the gamma function provides a continuous interpolation of the factorials, offset by one, the digamma function provides a continuous interpolation of the harmonic numbers , offset by the Euler–Mascheroni constant . The factorial function
1694-453: A profit for the state. Although the English probably first experimented with raffles and similar games of chance, the first recorded official lottery was chartered by Queen Elizabeth I , in the year 1566, and was drawn in 1569. The 400,000 tickets issued cost 10 shillings (£0.50) each (roughly three weeks of wages for ordinary citizens), with the grand prize worth roughly £5,000. This lottery
1815-404: A proof of Euclid's theorem that the number of primes is infinite. When n ! ± 1 {\displaystyle n!\pm 1} is itself prime it is called a factorial prime ; relatedly, Brocard's problem , also posed by Srinivasa Ramanujan , concerns the existence of square numbers of the form n ! + 1 {\displaystyle n!+1} . In contrast,
1936-525: A record for the largest lottery jackpot in U.S. history, with its 8 November 2022 draw having an estimated jackpot of US$ 2 billion. Factorial In mathematics , the factorial of a non-negative integer n {\displaystyle n} , denoted by n ! {\displaystyle n!} , is the product of all positive integers less than or equal to n {\displaystyle n} . The factorial of n {\displaystyle n} also equals
2057-422: A sequence. Factorials appear more broadly in many formulas in combinatorics , to account for different orderings of objects. For instance the binomial coefficients ( n k ) {\displaystyle {\tbinom {n}{k}}} count the k {\displaystyle k} -element combinations (subsets of k {\displaystyle k} elements) from
2178-399: A set with n {\displaystyle n} elements, and can be computed from factorials using the formula ( n k ) = n ! k ! ( n − k ) ! . {\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.} The Stirling numbers of the first kind sum to the factorials, and count
2299-418: A subset of exceptions with asymptotic density zero), it coincides with the largest prime factor of x {\displaystyle x} . The product of two factorials, m ! ⋅ n ! {\displaystyle m!\cdot n!} , always evenly divides ( m + n ) ! {\displaystyle (m+n)!} . There are infinitely many factorials that equal
2420-531: A way to raise public funding for projects, and this led to the popular belief that lotteries were a form of hidden tax. At the end of the Revolutionary War the various states had to resort to lotteries to raise funds for numerous public projects. The first big lottery on German soil was held in 1614 in Hamburg . In Austria the first lottery was drawn in 1751, during the reign of Empress Maria Theresia , and
2541-419: Is 1 {\displaystyle 1} , or in symbols, 0 ! = 1 {\displaystyle 0!=1} . There are several motivations for this definition: The earliest uses of the factorial function involve counting permutations : there are n ! {\displaystyle n!} different ways of arranging n {\displaystyle n} distinct objects into
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#17328522999142662-525: Is 1, according to the convention for an empty product . Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature , and by Jewish mystics in the Talmudic book Sefer Yetzirah . The factorial operation is encountered in many areas of mathematics, notably in combinatorics , where its most basic use counts the possible distinct sequences –
2783-518: Is a Malaysian company, which operates in the gambling sector. Founded and incorporated by the Malaysian Government in 1969, it was focused on the commercialisation of 4-Digits –based games. On 1 August 1985, the government in a non- tender privatisation , sold the company to businessman Vincent Tan who merged it into his Berjaya Group . Today, Sports Toto is a wholly owned subsidiary of Berjaya Sports Toto Berhad ( MYX : 1562), which
2904-565: Is a common feature in scientific calculators . It is also included in scientific programming libraries such as the Python mathematical functions module and the Boost C++ library . If efficiency is not a concern, computing factorials is trivial: just successively multiply a variable initialized to 1 {\displaystyle 1} by the integers up to n {\displaystyle n} . The simplicity of this computation makes it
3025-599: Is a national lottery. It is organized every year since 1812 by a branch of the Spanish Public Administration, now called Loterías y Apuestas del Estado. The name Sorteo de Navidad was used for the first time in 1892. The Spanish Christmas lottery is the second longest continuously running lottery in the world. This includes the years during the Spanish Civil War when the lottery draw was held in Valencia after
3146-550: Is a single multiplication of a number with O ( n log n ) {\displaystyle O(n\log n)} bits. Again, at each level of recursion the numbers involved have a constant fraction as many bits (because otherwise repeatedly squaring them would produce too large a final result) so again the amounts of time for these steps in the recursive calls add in a geometric series to O ( n log 2 n ) {\displaystyle O(n\log ^{2}n)} . Consequentially,
3267-414: Is calculated as n ! r ! ( n − r ) ! {\displaystyle n! \over r!(n-r)!} , where X! is notation for X factorial . Spreadsheets have the function COMBIN(n,r) to calculate ( n r ) {\displaystyle {n \choose r}} . To calculate "odds-to-1", divide the probability into 1.0 and subtract 1 from
3388-663: Is changed to keep all but the last term, it would define a product of the same form, for a smaller factorial. This leads to a recurrence relation , according to which each value of the factorial function can be obtained by multiplying the previous value by n {\displaystyle n} : n ! = n ⋅ ( n − 1 ) ! . {\displaystyle n!=n\cdot (n-1)!.} For example, 5 ! = 5 ⋅ 4 ! = 5 ⋅ 24 = 120 {\displaystyle 5!=5\cdot 4!=5\cdot 24=120} . The factorial of 0 {\displaystyle 0}
3509-558: Is divisible by n {\displaystyle n} if and only if n {\displaystyle n} is a prime number . For any given integer x {\displaystyle x} , the Kempner function of x {\displaystyle x} is given by the smallest n {\displaystyle n} for which x {\displaystyle x} divides n ! {\displaystyle n!} . For almost all numbers (all but
3630-622: Is listed on the main market of Bursa Malaysia . It claims to be the largest operator in Malaysia of 4D-based games, with 680 sales outlets offering a total of 7 games. The Mexican Lotería Nacional dates back to the late 18th century. The goal of the Lotería is to create jobs and to "impulse the wealth redistribution process". The Lotería is also a member of the North American Association of State and Provincial Lotteries. As measured by
3751-400: Is not uncommon to see casinos paying $ 500 or even $ 1,000 for a “catch” of 0 out of 20 on a 20 spot ticket with a $ 5.00 wager. Payouts vary widely by casino. Most casinos allow paytable wagers of 1 through 20 numbers, but some limit the choice to only 1 through 10, 12 and 15 numbers, or "spots" as keno aficionados call the numbers selected. The probability of a player hitting all 20 numbers on
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3872-406: Is operated by Allwyn Entertainment Ltd, who took over from the original operator Camelot Group in 2024. 28% of National Lottery revenue, along with all unclaimed prizes, are distributed as grants to charitable causes. 12% of the revenue from the National Lottery is expected to go to the government, 5% goes to lottery retailers, 5% is retained by the operator for operating costs, and 50% remains for
3993-440: Is positive. It can be extended to the non-integer points in the rest of the complex plane by solving for Euler's reflection formula Γ ( z ) Γ ( 1 − z ) = π sin π z . {\displaystyle \Gamma (z)\Gamma (1-z)={\frac {\pi }{\sin \pi z}}.} However, this formula cannot be used at integers because, for them,
4114-536: Is to perform the multiplications as a divide-and-conquer algorithm that multiplies a sequence of i {\displaystyle i} numbers by splitting it into two subsequences of i / 2 {\displaystyle i/2} numbers, multiplies each subsequence, and combines the results with one last multiplication. This approach to the factorial takes total time O ( n log 3 n ) {\displaystyle O(n\log ^{3}n)} : one logarithm comes from
4235-967: The ∼ {\displaystyle \sim } symbol means that, as n {\displaystyle n} goes to infinity, the ratio between the left and right sides approaches one in the limit . Stirling's formula provides the first term in an asymptotic series that becomes even more accurate when taken to greater numbers of terms: n ! ∼ 2 π n ( n e ) n ( 1 + 1 12 n + 1 288 n 2 − 139 51840 n 3 − 571 2488320 n 4 + ⋯ ) . {\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+{\frac {1}{12n}}+{\frac {1}{288n^{2}}}-{\frac {139}{51840n^{3}}}-{\frac {571}{2488320n^{4}}}+\cdots \right).} An alternative version uses only odd exponents in
4356-416: The sin π z {\displaystyle \sin \pi z} term would produce a division by zero . The result of this extension process is an analytic function , the analytic continuation of the integral formula for the gamma function. It has a nonzero value at all complex numbers, except for the non-positive integers where it has simple poles . Correspondingly, this provides
4477-419: The abc conjecture that there are only finitely many nontrivial examples. The greatest common divisor of the values of a primitive polynomial of degree d {\displaystyle d} over the integers evenly divides d ! {\displaystyle d!} . There are infinitely many ways to extend the factorials to a continuous function . The most widely used of these uses
4598-658: The Gambling Act 2005 , the maximum amount which can be won by a single ticket is £500,000, or 10% of the total draw proceeds. A minimum of 33% of the ticket price from players' subscriptions supports various trusts, which in turn fund local and international charities and community projects. Some £850 million have been donated. People's Postcode Lottery has a number of celebrity ambassadors, including David Attenborough , Judi Dench , Shobna Gulati , Tim Healy , Stephen Jardine , Ellen MacArthur , Aggie MacKenzie , Carey Mulligan , John Stapleton and Emma Thompson . The lottery
4719-569: The Great Wall of China . In modern China, the idea of using lotteries to fund a public institution was not accepted before the late 19th century. Chinese lottery is not documented before 1847, when the Portuguese government of Macao decided to grant a licence to lottery operators. According to some, results of keno games in great cities were sent to outlying villages and hamlets by carrier pigeons , resulting in its Chinese name 白鸽票 báigē piào , with
4840-528: The Sackur–Tetrode equation must correct the count of microstates by dividing by the factorials of the numbers of each type of indistinguishable particle to avoid the Gibbs paradox . Quantum physics provides the underlying reason for why these corrections are necessary. As a function of n {\displaystyle n} , the factorial has faster than exponential growth , but grows more slowly than
4961-542: The United States and some other countries during the 19th century, by the beginning of the 20th century, most forms of gambling, including lotteries and sweepstakes , were illegal in the U.S. and most of Europe as well as many other countries. This remained so until well after World War II. In the 1960s, casinos and lotteries began to re-appear throughout the world as a means for governments to raise revenue without raising taxes. Lotteries come in many formats. For example,
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5082-508: The Virginia Company of London the right to raise money to help establish settlers in the first permanent English colony at Jamestown, Virginia . Lotteries in colonial America played a significant part in the financing of both private and public ventures. It has been recorded that more than 200 lotteries were sanctioned between 1744 and 1776, and played a major role in financing roads, libraries, churches, colleges, canals, bridges, etc. In
5203-467: The Wallis product , which expresses π {\displaystyle \pi } as a limiting ratio of factorials and powers of two. The result of these corrections is Stirling's approximation : n ! ∼ 2 π n ( n e ) n . {\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\,.} Here,
5324-694: The Western Canada Lottery Corporation (which serves Western and Northern Canada , excluding British Columbia ), and the British Columbia Lottery Corporation . The five regional lotteries are members of a consortium known as the Interprovincial Lottery Corporation , which administrates national games, including the flagship Lotto 6/49 and Lotto Max . The five lotteries offer draw games, scratch cards, and sports betting —the latter primarily under
5445-465: The exponential generating function , which for a combinatorial class with n i {\displaystyle n_{i}} elements of size i {\displaystyle i} is defined as the power series ∑ i = 0 ∞ x i n i i ! . {\displaystyle \sum _{i=0}^{\infty }{\frac {x^{i}n_{i}}{i!}}.} In number theory ,
5566-459: The gamma function , which can be defined for positive real numbers as the integral Γ ( z ) = ∫ 0 ∞ x z − 1 e − x d x . {\displaystyle \Gamma (z)=\int _{0}^{\infty }x^{z-1}e^{-x}\,dx.} The resulting function is related to the factorial of a non-negative integer n {\displaystyle n} by
5687-579: The gamma function . Adrien-Marie Legendre included Legendre's formula , describing the exponents in the factorization of factorials into prime powers , in an 1808 text on number theory . The notation n ! {\displaystyle n!} for factorials was introduced by the French mathematician Christian Kramp in 1808. Many other notations have also been used. Another later notation | n _ {\displaystyle \vert \!{\underline {\,n}}} , in which
5808-408: The permutations – of n {\displaystyle n} distinct objects: there are n ! {\displaystyle n!} . In mathematical analysis , factorials are used in power series for the exponential function and other functions, and they also have applications in algebra , number theory , probability theory , and computer science . Much of the mathematics of
5929-406: The prime number theorem , so the time for the first step is O ( n log 2 n ) {\displaystyle O(n\log ^{2}n)} , with one logarithm coming from the divide and conquer and another coming from the multiplication algorithm. In the recursive calls to the algorithm, the prime number theorem can again be invoked to prove that the numbers of bits in
6050-694: The 1740s, the foundation of Princeton and Columbia Universities was financed by lotteries, as was the University of Pennsylvania by the Academy Lottery in 1755. During the French and Indian Wars , several colonies used lotteries to help finance fortifications and their local militia. In May 1758, the Province of Massachusetts Bay raised money with a lottery for the "Expedition against Canada". Benjamin Franklin organized
6171-781: The 2011 round 41. The main victory at that time was with 7 correct results and the smallest victory with three actual and one additional number, the number of which was reduced from three to two. The lottery return percentage is 41.1. Another lottery game played in Finland is Vikinglotto , which can be played in all Nordic countries as well as in Estonia , Latvia and Lithuania . In Vikinglotto, six actual numbers and two additional numbers out of 48 are drawn. There are five winning categories: 6 correct, 5 + extra number, 5 correct, 4 correct and 3 correct. In Finland, an average of six million euros in winnings go unredeemed each year. A national lottery
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#17328522999146292-661: The Australian government started their own lottery, named the 'Golden Casket Art Union', with the intention of raising money for charities and projects. Its first draw is credited with raising funds for veterans of World War One. Lotteries in Canada are administered by five regional organizations; the Atlantic Lottery Corporation (which serves Atlantic Canada ), Loto-Québec , the Ontario Lottery and Gaming Corporation ,
6413-494: The Calzabigi brothers (Giovanni and Ranieri). Casanova defended the project in a series of conversations with Madame de Pompadour , the French mathematician Jean d'Alembert , Joseph de Pâris Duverney, intendent of the École, and the French minister of foreign affair. Unlike modern lotteries where the state can never lose, in the French lottery the state could lose, but a wise choice of the payoff made losses so improbable as to ensure
6534-453: The English lotteries ran for over 250 years, until the government, under constant pressure from the opposition in Parliament, declared a final lottery in 1826. This lottery was held up to ridicule by contemporary commentators as "the last struggle of the speculators on public credulity for popularity to their last dying lottery". An English lottery, authorized by King James I in 1612, granted
6655-702: The Republicans were forced to relocate their capital from Madrid. After the overthrow of the Republican government the lottery continued uninterrupted under the Franco regime. Notable prizes on different continents are: (local currency) The first lottery in Australia took place in the 1880s in Sydney. It was a private sweepstakes that was quickly prohibited, despite being moved to other areas such as Queensland and Victoria. In 1916,
6776-418: The analysis of brute-force searches over permutations, factorials arise in the lower bound of log 2 n ! = n log 2 n − O ( n ) {\displaystyle \log _{2}n!=n\log _{2}n-O(n)} on the number of comparisons needed to comparison sort a set of n {\displaystyle n} items, and in
6897-403: The analysis of chained hash tables , where the distribution of keys per cell can be accurately approximated by a Poisson distribution. Moreover, factorials naturally appear in formulae from quantum and statistical physics , where one often considers all the possible permutations of a set of particles. In statistical mechanics , calculations of entropy such as Boltzmann's entropy formula or
7018-542: The argument of the factorial was half-enclosed by the left and bottom sides of a box, was popular for some time in Britain and America but fell out of use, perhaps because it is difficult to typeset. The word "factorial" (originally French: factorielle ) was first used in 1800 by Louis François Antoine Arbogast , in the first work on Faà di Bruno's formula , but referring to a more general concept of products of arithmetic progressions . The "factors" that this name refers to are
7139-610: The brand Sport Select . The largest single jackpot record in Canadian lottery history was a Lotto Max drawing on January 7, 2020, for a jackpot of $ 70 million. In Finland, Veikkaus began selling lottery tickets in December 1970, and the first draw was televised on January 3, 1971. The lottery turned 40 on January 3, 2011, and by then the lottery had been drawn 2,126 times. Since then, there has been one lottery draw every week. Lottery game time usually ends on Saturday at 9:45 p.m., and
7260-432: The coefficients of other Taylor series (in particular those of the trigonometric and hyperbolic functions ), where they cancel factors of n ! {\displaystyle n!} coming from the n {\displaystyle n} th derivative of x n {\displaystyle x^{n}} . This usage of factorials in power series connects back to analytic combinatorics through
7381-400: The complex gamma function and its scalar multiples are the only holomorphic functions on the positive complex half-plane that obey the functional equation and remain bounded for complex numbers with real part between 1 and 2. Other complex functions that interpolate the factorial values include Hadamard's gamma function , which is an entire function over all the complex numbers, including
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#17328522999147502-721: The correction terms: n ! ∼ 2 π n ( n e ) n exp ( 1 12 n − 1 360 n 3 + 1 1260 n 5 − 1 1680 n 7 + ⋯ ) . {\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\exp \left({\frac {1}{12n}}-{\frac {1}{360n^{3}}}+{\frac {1}{1260n^{5}}}-{\frac {1}{1680n^{7}}}+\cdots \right).} Many other variations of these formulas have also been developed, by Srinivasa Ramanujan , Bill Gosper , and others. The binary logarithm of
7623-515: The corresponding products decrease by a constant factor at each level of recursion, so the total time for these steps at all levels of recursion adds in a geometric series to O ( n log 2 n ) {\displaystyle O(n\log ^{2}n)} . The time for the squaring in the second step and the multiplication in the third step are again O ( n log 2 n ) {\displaystyle O(n\log ^{2}n)} , because each
7744-496: The denominators of power series , most notably in the series for the exponential function , e x = 1 + x 1 + x 2 2 + x 3 6 + ⋯ = ∑ i = 0 ∞ x i i ! , {\displaystyle e^{x}=1+{\frac {x}{1}}+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+\cdots =\sum _{i=0}^{\infty }{\frac {x^{i}}{i!}},} and in
7865-470: The draw is usually held on Saturday at 10:15 p.m. Large public holidays on Saturdays may postpone the draw to Sunday. The lottery has two official supervisors; from 3 January 1971 to 29 September 2013, the lottery was televised on Yle TV1 , and in October 2013, the lottery draws were postponed on MTV3 after ten evening news, because according to FICORA , the sponsorship cooperation between Veikkaus and Yle
7986-557: The drawing of lots. The first known European lotteries were held during the Roman Empire , mainly as an amusement at dinner parties. Each guest would receive a ticket, and prizes would often consist of fancy items such as dinnerware. Every ticket holder would be assured of winning something. This type of lottery, however, was no more than the distribution of gifts by wealthy noblemen during the Saturnalian revelries. The earliest records of
8107-471: The equation n ! = Γ ( n + 1 ) , {\displaystyle n!=\Gamma (n+1),} which can be used as a definition of the factorial for non-integer arguments. At all values x {\displaystyle x} for which both Γ ( x ) {\displaystyle \Gamma (x)} and Γ ( x − 1 ) {\displaystyle \Gamma (x-1)} are defined,
8228-420: The factorial function are commonly used as an example of different computer programming styles, and are included in scientific calculators and scientific computing software libraries. Although directly computing large factorials using the product formula or recurrence is not efficient, faster algorithms are known, matching to within a constant factor the time for fast multiplication algorithms for numbers with
8349-407: The factorial function was developed beginning in the late 18th and early 19th centuries. Stirling's approximation provides an accurate approximation to the factorial of large numbers, showing that it grows more quickly than exponential growth . Legendre's formula describes the exponents of the prime numbers in a prime factorization of the factorials, and can be used to count the trailing zeros of
8470-907: The factorial implies that n ! {\displaystyle n!} is divisible by all prime numbers that are at most n {\displaystyle n} , and by no larger prime numbers. More precise information about its divisibility is given by Legendre's formula , which gives the exponent of each prime p {\displaystyle p} in the prime factorization of n ! {\displaystyle n!} as ∑ i = 1 ∞ ⌊ n p i ⌋ = n − s p ( n ) p − 1 . {\displaystyle \sum _{i=1}^{\infty }\left\lfloor {\frac {n}{p^{i}}}\right\rfloor ={\frac {n-s_{p}(n)}{p-1}}.} Here s p ( n ) {\displaystyle s_{p}(n)} denotes
8591-722: The factorial involves repeated products, rather than a single multiplication, so these time bounds do not apply directly. In this setting, computing n ! {\displaystyle n!} by multiplying the numbers from 1 to n {\displaystyle n} in sequence is inefficient, because it involves n {\displaystyle n} multiplications, a constant fraction of which take time O ( n log 2 n ) {\displaystyle O(n\log ^{2}n)} each, giving total time O ( n 2 log 2 n ) {\displaystyle O(n^{2}\log ^{2}n)} . A better approach
8712-620: The factorial, used to analyze comparison sorting , can be very accurately estimated using Stirling's approximation. In the formula below, the O ( 1 ) {\displaystyle O(1)} term invokes big O notation . log 2 n ! = n log 2 n − ( log 2 e ) n + 1 2 log 2 n + O ( 1 ) . {\displaystyle \log _{2}n!=n\log _{2}n-(\log _{2}e)n+{\frac {1}{2}}\log _{2}n+O(1).} The product formula for
8833-658: The factorials arise through the binomial theorem , which uses binomial coefficients to expand powers of sums. They also occur in the coefficients used to relate certain families of polynomials to each other, for instance in Newton's identities for symmetric polynomials . Their use in counting permutations can also be restated algebraically: the factorials are the orders of finite symmetric groups . In calculus , factorials occur in Faà di Bruno's formula for chaining higher derivatives. In mathematical analysis , factorials frequently appear in
8954-431: The factorials. Daniel Bernoulli and Leonhard Euler interpolated the factorial function to a continuous function of complex numbers , except at the negative integers, the (offset) gamma function . Many other notable functions and number sequences are closely related to the factorials, including the binomial coefficients , double factorials , falling factorials , primorials , and subfactorials . Implementations of
9075-401: The factorization of a binomial coefficient. Grouping the prime factors of the factorial into prime powers in different ways produces the multiplicative partitions of factorials . The special case of Legendre's formula for p = 5 {\displaystyle p=5} gives the number of trailing zeros in the decimal representation of the factorials. According to this formula,
9196-466: The first results of Paul Erdős , was based on the divisibility properties of factorials. The factorial number system is a mixed radix notation for numbers in which the place values of each digit are factorials. Factorials are used extensively in probability theory , for instance in the Poisson distribution and in the probabilities of random permutations . In computer science , beyond appearing in
9317-427: The gamma function obeys the functional equation Γ ( n ) = ( n − 1 ) Γ ( n − 1 ) , {\displaystyle \Gamma (n)=(n-1)\Gamma (n-1),} generalizing the recurrence relation for the factorials. The same integral converges more generally for any complex number z {\displaystyle z} whose real part
9438-739: The literal reading "white dove tickets" in Mandarin, but in Southern varieties of Chinese spoken in Guangdong simply meaning "pigeon tickets", and pronounced baak-gaap-piu in Cantonese (on which the Western spelling 'pak-ah-pu' / ' pakapoo ' was based). The Chinese played the game using sheets printed with Chinese characters , often the first 80 characters of the Thousand Character Classic , from which
9559-694: The lottery's structure was designed to contravene British law regarding lotteries. In the UK, winning the lottery is correlated to expressing more preference for the Conservative Party . Winning larger prizes results in a larger shift in favor of the Conservative Party. People's Postcode Lottery is a subscription lottery in the UK. The format was introduced by Dutch company Novamedia BV : players pay at minimum £10 monthly to play, and winning postcodes are announced daily. In accordance with restrictions under
9680-402: The modern game of Keno. Keno payouts are based on how many numbers the player chooses and how many of those numbers are "hit", multiplied by the proportion of the player's original wager to the "base rate" of the paytable. Typically, the more numbers a player chooses and the more numbers hit, the greater the payout, although some paytables pay for hitting a lesser number of spots. For example, it
9801-520: The modern-day stockbrokers for various commercial ventures. Most people could not afford the entire cost of a lottery ticket, so the brokers would sell shares in a ticket; this resulted in tickets being issued with a notation such as "Sixteenth" or "Third Class". Many private lotteries were held, including raising money for the Virginia Company of London to support its settlement in America at Jamestown. The English State Lottery ran from 1694 until 1826. Thus,
9922-449: The most salient property of factorials is the divisibility of n ! {\displaystyle n!} by all positive integers up to n {\displaystyle n} , described more precisely for prime factors by Legendre's formula . It follows that arbitrarily large prime numbers can be found as the prime factors of the numbers n ! ± 1 {\displaystyle n!\pm 1} , leading to
10043-423: The non-positive integers. In the p -adic numbers , it is not possible to continuously interpolate the factorial function directly, because the factorials of large integers (a dense subset of the p -adics) converge to zero according to Legendre's formula, forcing any continuous function that is close to their values to be zero everywhere. Instead, the p -adic gamma function provides a continuous interpolation of
10164-407: The number of bits in the factorial, a second comes from the multiplication algorithm, and a third comes from the divide and conquer. Even better efficiency is obtained by computing n ! from its prime factorization, based on the principle that exponentiation by squaring is faster than expanding an exponent into a product. An algorithm for this by Arnold Schönhage begins by finding the list of
10285-498: The number of spots and numbers that are picked on each ticket. Keno probabilities come from a hypergeometric distribution . For Keno, one calculates the probability of hitting exactly r {\displaystyle r} spots on an n {\displaystyle n} -spot ticket by the formula: To calculate the probability of hitting 4 spots on a 6-spot ticket, the formula is: where ( n r ) {\displaystyle {n \choose r}}
10406-460: The number of zeros can be obtained by subtracting the base-5 digits of n {\displaystyle n} from n {\displaystyle n} , and dividing the result by four. Legendre's formula implies that the exponent of the prime p = 2 {\displaystyle p=2} is always larger than the exponent for p = 5 {\displaystyle p=5} , so each factor of five can be paired with
10527-457: The number of €20 tickets available was reduced from 180 million to 160 million, reducing the potential maximum prize pool to €2.24 billion (70% of ticket sales), with a maximum potential El Gordo of €720 million. A lottery was first held in Thailand (then known as Siam ) in 1874 during the reign of King Chulalongkorn (Rama V), as part of an international fair organised for his birthday. A lottery
10648-423: The numbers n ! + 2 , n ! + 3 , … n ! + n {\displaystyle n!+2,n!+3,\dots n!+n} must all be composite, proving the existence of arbitrarily large prime gaps . An elementary proof of Bertrand's postulate on the existence of a prime in any interval of the form [ n , 2 n ] {\displaystyle [n,2n]} , one of
10769-597: The outset of the Revolutionary War, the Continental Congress used lotteries to raise money to support the Colonial Army. Alexander Hamilton wrote that lotteries should be kept simple, and that "Everybody ... will be willing to hazard a trifling sum for the chance of considerable gain ... and would prefer a small chance of winning a great deal to a great chance of winning little". Taxes had never been accepted as
10890-454: The permutations of n {\displaystyle n} grouped into subsets with the same numbers of cycles. Another combinatorial application is in counting derangements , permutations that do not leave any element in its original position; the number of derangements of n {\displaystyle n} items is the nearest integer to n ! / e {\displaystyle n!/e} . In algebra ,
11011-601: The poor or in order to raise funds for a wide range of public usages. The lotteries proved very popular and were hailed as a painless form of taxation . The Dutch state-owned Staatsloterij is the oldest running lottery (1726). The English word lottery is derived from the Dutch noun "lot" meaning "fate". The first recorded Italian lottery was held on 9 January 1449 in Milan organized by the Golden Ambrosian Republic to finance
11132-471: The poor. The town records of Ghent , Utrecht , and Bruges indicate that lotteries may be even older. A record dated 9 May 1445 at L'Ecluse refers to raising funds to build walls and town fortifications, with a lottery of 4,304 tickets and total prize money of 1737 florins (worth about US$ 170,000 in 2014). In the 17th century it was quite usual in the Netherlands to organize lotteries to collect money for
11253-528: The possibility of multiple winners. The first recorded signs of a lottery are keno slips from the Chinese Han dynasty between 205 and 187 BC. These lotteries are believed to have helped to finance major government projects like the Great Wall of China . From the Chinese Book of Songs (2nd millennium BC.) comes a reference to a game of chance as "the drawing of wood", which in context appears to describe
11374-458: The primes up to n {\displaystyle n} , for instance using the sieve of Eratosthenes , and uses Legendre's formula to compute the exponent for each prime. Then it computes the product of the prime powers with these exponents, using a recursive algorithm, as follows: The product of all primes up to n {\displaystyle n} is an O ( n ) {\displaystyle O(n)} -bit number, by
11495-418: The prize can be a fixed amount of cash or goods. In this format, there is risk to the organizer if insufficient tickets are sold. More commonly, the prize fund will be a fixed percentage of the receipts. A popular form of this is the "50–50" draw, where the organizers promise that the prize will be 50% of the revenue. Many recent lotteries allow purchasers to select the numbers on the lottery ticket, resulting in
11616-864: The product of n {\displaystyle n} with the next smaller factorial: n ! = n × ( n − 1 ) × ( n − 2 ) × ( n − 3 ) × ⋯ × 3 × 2 × 1 = n × ( n − 1 ) ! {\displaystyle {\begin{aligned}n!&=n\times (n-1)\times (n-2)\times (n-3)\times \cdots \times 3\times 2\times 1\\&=n\times (n-1)!\\\end{aligned}}} For example, 5 ! = 5 × 4 ! = 5 × 4 × 3 × 2 × 1 = 120. {\displaystyle 5!=5\times 4!=5\times 4\times 3\times 2\times 1=120.} The value of 0!
11737-949: The product of other factorials: if n {\displaystyle n} is itself any product of factorials, then n ! {\displaystyle n!} equals that same product multiplied by one more factorial, ( n − 1 ) ! {\displaystyle (n-1)!} . The only known examples of factorials that are products of other factorials but are not of this "trivial" form are 9 ! = 7 ! ⋅ 3 ! ⋅ 3 ! ⋅ 2 ! {\displaystyle 9!=7!\cdot 3!\cdot 3!\cdot 2!} , 10 ! = 7 ! ⋅ 6 ! = 7 ! ⋅ 5 ! ⋅ 3 ! {\displaystyle 10!=7!\cdot 6!=7!\cdot 5!\cdot 3!} , and 16 ! = 14 ! ⋅ 5 ! ⋅ 2 ! {\displaystyle 16!=14!\cdot 5!\cdot 2!} . It would follow from
11858-782: The recursive version takes linear space to store its call stack . However, this model of computation is only suitable when n {\displaystyle n} is small enough to allow n ! {\displaystyle n!} to fit into a machine word . The values 12! and 20! are the largest factorials that can be stored in, respectively, the 32-bit and 64-bit integers . Floating point can represent larger factorials, but approximately rather than exactly, and will still overflow for factorials larger than 170 ! {\displaystyle 170!} . The exact computation of larger factorials involves arbitrary-precision arithmetic , because of fast growth and integer overflow . Time of computation can be analyzed as
11979-560: The result (and ignoring the negligible + 1 {\displaystyle +1} term) approximates n ! {\displaystyle n!} as ( n / e ) n {\displaystyle (n/e)^{n}} . More carefully bounding the sum both above and below by an integral, using the trapezoid rule , shows that this estimate needs a correction factor proportional to n {\displaystyle {\sqrt {n}}} . The constant of proportionality for this correction can be found from
12100-459: The result. Lottery A lottery (or lotto ) is a form of gambling that involves the drawing of numbers at random for a prize. Some governments outlaw lotteries, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree of regulation of lottery by governments. The most common regulations are prohibition of sale to minors and licensing of ticket vendors. Although lotteries were common in
12221-417: The same number of digits. The concept of factorials has arisen independently in many cultures: From the late 15th century onward, factorials became the subject of study by Western mathematicians. In a 1494 treatise, Italian mathematician Luca Pacioli calculated factorials up to 11!, in connection with a problem of dining table arrangements. Christopher Clavius discussed factorials in a 1603 commentary on
12342-420: The sum of the base - p {\displaystyle p} digits of n {\displaystyle n} , and the exponent given by this formula can also be interpreted in advanced mathematics as the p -adic valuation of the factorial. Applying Legendre's formula to the product formula for binomial coefficients produces Kummer's theorem , a similar result on the exponent of each prime in
12463-736: The terms of the product formula for the factorial. The factorial function of a positive integer n {\displaystyle n} is defined by the product of all positive integers not greater than n {\displaystyle n} n ! = 1 ⋅ 2 ⋅ 3 ⋯ ( n − 2 ) ⋅ ( n − 1 ) ⋅ n . {\displaystyle n!=1\cdot 2\cdot 3\cdots (n-2)\cdot (n-1)\cdot n.} This may be written more concisely in product notation as n ! = ∏ i = 1 n i . {\displaystyle n!=\prod _{i=1}^{n}i.} If this product formula
12584-641: The top price now up to 32 million baht . Shrines of local folklore and popular religion , such as Nang Ta-Khian , are often propitiated in order to be lucky in the Thai lottery draw. The principal lottery in the United Kingdom is the National Lottery , a state-franchised lottery sanctioned by the Gambling Commission (formerly the National Lottery Commission ), and established in 1994. It
12705-486: The total prize fund of which 5% is diverted to a Super Draw fund, leaving 45% for normal prizes. Northern & Shell also operates a commercial lottery known as The Health Lottery , which distributes its revenue to support health-related charities and causes. To comply with the Gambling Act, which forbids other parties from operating a national lottery, The Health Lottery operates as an umbrella corporation representing
12826-528: The total prize payout, the Spanish Christmas Lottery is considered the biggest lottery worldwide. In 2012, if all of the tickets had been sold, the total amount payout of prizes would have been worth €2.52 billion (70% of ticket sales). The total amount of all prizes of the first category called El Gordo ("the fat one") was €720 million which was distributed among 180 winning tickets (billetes) that win €4 million each. For 2013, due to falling demand,
12947-518: The typical house edge for non-slot casino games is under 5%. The word "keno" has French or Latin roots (Fr. quine "five winning numbers", L. quini "five each"), but by all accounts the game originated in China. Legend has it that Zhang Liang invented the game during the Chu-Han Contention to raise money to defend an ancient city, and its widespread popularity later helped raise funds to build
13068-480: The unit-cost random-access machine model of computation, in which each arithmetic operation takes constant time and each number uses a constant amount of storage space. In this model, these methods can compute n ! {\displaystyle n!} in time O ( n ) {\displaystyle O(n)} , and the iterative version uses space O ( 1 ) {\displaystyle O(1)} . Unless optimized for tail recursion ,
13189-528: The war against the Republic of Venice . However, it was in Genoa that Lotto became very popular. People used to bet on the name of Great Council members, who were drawn by chance , five out of ninety candidates every six months. This kind of gambling was called Lotto or Semenaiu . When people wanted to bet more frequently than twice a year, they began to substitute the candidates names with numbers and modern lotto
13310-567: The winning characters were selected. Eventually, Chinese immigrants introduced keno to the West when they sailed across the Pacific Ocean to work on construction of the First transcontinental railroad in the 19th century, where the name was Westernized into boc hop bu and puck-apu . There were also other, earlier games called Keno, but these were played in the same way as the game now known as " Bingo ", not
13431-542: The work of Johannes de Sacrobosco , and in the 1640s, French polymath Marin Mersenne published large (but not entirely correct) tables of factorials, up to 64!, based on the work of Clavius. The power series for the exponential function , with the reciprocals of factorials for its coefficients, was first formulated in 1676 by Isaac Newton in a letter to Gottfried Wilhelm Leibniz . Other important works of early European mathematics on factorials include extensive coverage in
13552-641: The world thanks to online lottery agents and bookkeepers. Some states have tried to combat this with different measures. The state government of Tamil Nadu decided to ban GooglePay since it allows payments to online lotteries and awards its users in India with Scratchcards . Indian lotteries provide a substantial economic boost for the states that provide them. In the fiscal year 2017–2018 Kerala collected GST worth Rs 908 crore and state revenue of Rs 1,691 crore. Lottery industry start operated in Malaysia on early 1969 by Berjaya Group . Sports Toto Malaysia Sdn Bhd
13673-533: Was a fiasco, since the tickets were very costly and the social classes which could afford them opposed the project. Between 1757 and 1836, for a period of about 80 years with some interruption during the French Revolution , the French state ran a profitable Loterie . The project was born out a series of initiatives to fund the École militaire . Instrumental to the birth of the Loterie were Giacomo Casanova and
13794-507: Was born, to which both modern legal lotteries and the illegal numbers game can trace their ancestry. King Francis I of France discovered the lotteries during his campaigns in Italy and decided to organize such a lottery in his kingdom to help the state finances. The first French lottery, the Loterie Royale , was held in 1539 and was authorized with the edict of Châteaurenard . This attempt
13915-505: Was designed to raise money for the "reparation of the havens and strength of the Realme, and towardes such other publique good workes", including the rebuilding of ports and new ships for the royal fleet. Each ticket holder won a prize, and the total value of the prizes equalled the money raised. Prizes were in the form of both "ready money" and valuable commodities such as silver plate, tapestries, and fine linen cloth. Additionally, each participant
14036-515: Was granted immunity from one arrest, "so long as the crime wasn't piracy, murder, felonies, or treason." The lottery was promoted by scrolls posted throughout the country showing sketches of the prizes. Thus, the lottery money received was an interest-free loan to the government during the three years that the tickets ('without any Blankes') were sold. In later years, the government sold the lottery ticket rights to brokers, who in turn hired agents and runners to sell them. These brokers eventually became
14157-458: Was illegal. In the current lottery played in Finland, the player chooses seven numbers between 1 and 40 (initially, until the autumn of 1980, six numbers between 1 and 40 were chosen, then for a few years seven numbers between 1 and 37 and then seven numbers between 1 and 39). In the draw, seven numbers and one (previously three and then two) additional numbers are drawn; the line price is 1 euro. The profit categories were changed, for example, from
14278-716: Was named Lotto di Genova since it was based on 90 numbers. Spain offers a wealth of lottery games, the majority of which are operated by Loterías y Apuestas del Estado with the remaining lotteries operated by the ONCE and the Catalan government. The first Spanish lottery game was played back in 1763 and, over the last two centuries, playing the lottery in Spain has developed into a tradition. The Spanish Christmas Lottery (officially Sorteo Extraordinario de Navidad [soɾˈteo ekstɾaorðiˈnaɾjo ðe naβiˈðað] or simply Lotería de Navidad [loteˈɾia ðe naβiˈðað])
14399-505: Was organised in 1917 by the British government with Thai consent to help finance Britain's war effort. Lotteries were held intermittently until 1933, when they became regularised under the finance department. The present Thai lottery is managed by The Government Lottery Office, a state enterprise managed by the Ministry of Finance . The drawings take place on the 1st and 16th of each month, with
14520-986: Was reintroduced in 1933. There are many lotteries in India. All lotteries are run by state governments but only 13 of the 28 Indian states allow them. The leader within Indian lotteries is the Kerala State Government that started their lottery department in 1967 following the country wide ban on private lotteries. The Kerala State Lotteries became an inspiration for other Indian states that started their own lotteries. As of right now lotteries are available in Kerala , Goa , Maharashtra , Madhya Pradesh , Punjab , West Bengal , Assam , Arunachal Pradesh , Meghalaya , Manipur , Sikkim , Nagaland and Mizoram . The public ban on lotteries in other states has not been very effective since several lottery providers allow Indians to play online. Indian players can play lotteries from all over
14641-523: Was the subject of a Channel 4 documentary, The Welsh Valley That Won the Lottery , about the residents of Rhymney who won in May 2022. Lotteries are operated at the state level in the U.S.; 45 states and 3 territories operate state lotteries, and nearly all of them are members of consortiums that operate regional games, and the two near-national games Mega Millions and Powerball . In November 2022, Powerball set
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