compute binomial coefficients
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 n2 .
If the arguments are both non-negative integers with 0≤r≤n, then nr=n!r!⁢n−r!, 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≤r≤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.
computing sequences of binomial coefficients is optimized to be faster than computing each one in isolation
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