GAP Manual: 46.2. Binomial

Binomial( n, k )

Binomial returns the binomial coefficient (n choosek) of integers n and k, which is defined as n! / (k! (n-k)!) (see Factorial). We define (0 choose0) = 1, (n choosek) = 0 if k<0 or n<k, and (n choosek) = (-1)^k (-n+k-1 choosek) if n < 0, which is consistent with (n choosek) = (n-1 choosek) + (n-1 choosek-1).

(n choosek) is the number of combinations with k elements, i.e., the number of subsets with k elements, of a set with n elements. (n choosek) is the coefficient of the term x^k of the polynomial (x + 1)^n, which is the generating function for (n choose*), hence the name.

    gap> List( [0..4], k->Binomial( 4, k ) );
    [ 1, 4, 6, 4, 1 ]    # Knuth calls this the trademark of Binomial
    gap> List( [0..6], n->List( [0..6], k->Binomial( n, k ) ) );;
    gap> PrintArray( last );
    [ [   1,   0,   0,   0,   0,   0,   0 ],    # the lower triangle is
      [   1,   1,   0,   0,   0,   0,   0 ],    # called Pascal\'s triangle
      [   1,   2,   1,   0,   0,   0,   0 ],
      [   1,   3,   3,   1,   0,   0,   0 ],
      [   1,   4,   6,   4,   1,   0,   0 ],
      [   1,   5,  10,  10,   5,   1,   0 ],
      [   1,   6,  15,  20,  15,   6,   1 ] ]
    gap> Binomial( 50, 10 );
    10272278170 

NrCombinations (see Combinations) is the generalization of Binomial for multisets. Combinations (see Combinations) computes the set of all combinations of a multiset.