The binomial(n,r) function computes binomial coefficients.

You can enter the command binomial using either the 1-D or 2-D calling sequence. For example, binomial(n, 2) is equivalent to $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ .

If the arguments are both non-negative integers with $0\le r\le n$, then $\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\frac{n!}{r!\left(n-r\right)!}$, which is the number of distinct sets of r objects that can be chosen from n distinct objects.

If n and r are integers that do not satisfy $0\le r\le n$, or $n$ and $r$ are rationals or floating-point numbers, then the general definition is used, that is,

At all points $n,r$ where none of $n$, $r$, and $n-r$ is a negative integer, the above definition is equivalent to:

In the case that $n$ is a negative integer, binomial(n,r) is defined by this limit. If $r$ is a negative integer, by the symmetry relation binomial(n,r) = binomial(n,n-r), the above limit is used.

In the case that exactly two of the expressions $n$, $r$, and $n-r$ are negative integers, Maple also signals the invalid_operation numeric event, allowing the user to control this singular behavior by catching the event. See numeric_events for more information.

For symbolic arguments, some simplifications, for example, binomial(n, 1) = n, can be made, but typically binomial returns unevaluated.

For positive integer arguments, binomial is computed using GMP. A limited number of previous computed values will be cached and new values will be computed using a recurrence formula.  In practice that means that it is very fast to compute sequences of binomial coefficients for fixed values of $n$ or $r$.

$\mathrm{binomial}\left(4\,2\right)$

$6$

$\mathrm{binomial}\left(2\,\frac{1}{2}\right)$

$\frac{16}{3\mathrm{\pi }}$

$\mathrm{binomial}\left(2.1\,2+3\mathrm{I}\right)$

$\mathrm{-56.56167619}-98.27156511\mathrm{I}$

$\mathrm{binomial}\left(n\,2\right)$

$\left(\genfrac{}{}{0ex}{}{n}{2}\right)$

$\mathrm{expand}\left(\right)$

$\frac{1}{2}{n}^2-\frac{1}{2}n$

$\mathrm{NumericStatus}\left(\mathrm{invalid\_operation}=\mathrm{false}\right)\:$

$\mathrm{binomial}\left(-3\,5\right)$

$\mathrm{-21}$

$\mathrm{NumericStatus}\left(\mathrm{invalid\_operation}\right)$

$\mathrm{true}$

$\mathrm{seq}\left(\mathrm{binomial}\left(100\,i\right)\,i=50..60\right)$

$100891344545564193334812497256,98913082887808032681188722800,93206558875049876949581681100,84413487283064039501507937600,73470998190814997343905056800,61448471214136179596720592960,49378235797073715747364762200,38116532895986727945334202400,28258808871162574166368460400,20116440213369968050635175200,13746234145802811501267369720$