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{{Built-in|Binomial|!}} is a [[dyadic]] [[scalar function]] which gives the [[wikipedia:binomial coefficient|binomial coefficient]] <math>\tbinom nk</math> between the two [[argument|arguments]]. The argument order <syntaxhighlight lang=apl inline>k!n</ | {{Built-in|Binomial|!}} is a [[dyadic]] [[scalar function]] which gives the [[wikipedia:binomial coefficient|binomial coefficient]] <math>\tbinom nk</math> between the two [[argument|arguments]]. The argument order <syntaxhighlight lang=apl inline>k!n</syntaxhighlight> is reversed compared to most of traditional mathematical notation's alternative notations, for example <math>C(n,k)</math> and <math>_nC_k</math>, but not others, like <math>C_n^k</math>. Binomial shares the [[glyph]] <syntaxhighlight lang=apl inline>!</syntaxhighlight> with the monadic arithmetic function [[Factorial]]. | ||
== Examples == | == Examples == | ||
For non-negative integer arguments, the binomial coefficient <syntaxhighlight lang=apl inline>k!n</ | For non-negative integer arguments, the binomial coefficient <syntaxhighlight lang=apl inline>k!n</syntaxhighlight> is equal to the number of ways to choose k items out of n distinct items. For example, <syntaxhighlight lang=apl inline>3!5</syntaxhighlight> is 10 because there are 10 ways to pick 3 items out of 5: 123, 124, 125, 134, 135, 145, 234, 235, 245, 345. | ||
<syntaxhighlight lang=apl> | <syntaxhighlight lang=apl> | ||
0 1 2 3 4 5!5 | 0 1 2 3 4 5!5 | ||
1 5 10 10 5 1 | 1 5 10 10 5 1 | ||
</ | </syntaxhighlight> | ||
<syntaxhighlight lang=apl inline>k!n</ | <syntaxhighlight lang=apl inline>k!n</syntaxhighlight> also corresponds to the k-th value (zero-indexed) on the n-th row (also zero-indexed) of [[wikipedia:Pascal's triangle|Pascal's triangle]]. | ||
<syntaxhighlight lang=apl> | <syntaxhighlight lang=apl> | ||
Line 20: | Line 20: | ||
1 4 6 4 1 0 | 1 4 6 4 1 0 | ||
1 5 10 10 5 1 | 1 5 10 10 5 1 | ||
</ | </syntaxhighlight>{{Works in|[[Dyalog APL]]}} | ||
== Properties == | == Properties == | ||
The value of <syntaxhighlight lang=apl inline>X!Y</ | The value of <syntaxhighlight lang=apl inline>X!Y</syntaxhighlight> equals <syntaxhighlight lang=apl inline>(!Y)÷(!X)×!Y-X</syntaxhighlight>. | ||
<syntaxhighlight lang=apl> | <syntaxhighlight lang=apl> | ||
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0 1 2 3 4 5 Alt 5 | 0 1 2 3 4 5 Alt 5 | ||
1 5 10 10 5 1 | 1 5 10 10 5 1 | ||
</ | </syntaxhighlight> | ||
In multiple implementations where [[Factorial]] is extended to use the [[wikipedia:Gamma function|Gamma function]] <math>\Gamma(n)</math>, Binomial is defined to use the above equality for non-integers. In that case, the [[wikipedia:Beta function|Beta function]] <math>\Beta(x,y)</math> becomes closely related to the Binomial, giving the identity <math>\Beta(X,Y)</math>{{←→}}<syntaxhighlight lang=apl inline>÷Y×(X-1)!X+Y-1</ | In multiple implementations where [[Factorial]] is extended to use the [[wikipedia:Gamma function|Gamma function]] <math>\Gamma(n)</math>, Binomial is defined to use the above equality for non-integers. In that case, the [[wikipedia:Beta function|Beta function]] <math>\Beta(x,y)</math> becomes closely related to the Binomial, giving the identity <math>\Beta(X,Y)</math>{{←→}}<syntaxhighlight lang=apl inline>÷Y×(X-1)!X+Y-1</syntaxhighlight>. | ||
<syntaxhighlight lang=apl> | <syntaxhighlight lang=apl> | ||
Line 39: | Line 39: | ||
2!3j2 | 2!3j2 | ||
1J5 | 1J5 | ||
</ | </syntaxhighlight>{{Works in|[[Dyalog APL]]}} | ||
== External links == | == External links == |