The numbcomp command counts the number of distinct ordered k-tuples of positive integers whose elements sum to n. For information on the composition command, which constructs k-compositions, see the combinat[composition] help page.

Note that $\mathrm{numbcomp}\left(n\,k\right)=\mathrm{nops}\left(\mathrm{composition}\left(n,k\right)\right)\=\mathrm{binomial}\left(n-1\,k-1\right)$ for $n,k>0$ .

$\mathrm{with}\left(\mathrm{combinat}\,\mathrm{numbcomp}\right)$

$\left[\mathrm{numbcomp}\right]$

$\mathrm{numbcomp}\left(7\,3\right)$

$15$

$\mathrm{numbcomp}\left(5\,4\right)$

$4$