If n is a list or set, then numbcomb counts the combinations of the elements of n taken m at a time. If m is not given, then all combinations are considered. If n is a non-negative integer, it is interpreted in the same way as a set of the first n integers.

Note that the result of numbcomb(n, m) is equivalent to $\mathrm{numelems}\left(\mathrm{choose}\left(n\,m\right)\right)$. However, this number is computed either by using binomial coefficients or by using a generating function method.

Additionally, note that if n is a non-negative integer, the result of numbcomb(n, m) is identical to that of $\left(\genfrac{}{}{0ex}{}{n}{m}\right)$.

The count of combinations takes into account duplicates in n. In the case where there are no duplicates, the count is given by the formula $2^n$ if m is not specified, or by the formula $\left(\genfrac{}{}{0ex}{}{n}{m}\right)$ if m is specified. If there are duplicates in the list, then the generating function is used.

The command with(combinat,numbcomb) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{combinat}\right)\:$

$\mathrm{numbcomb}\left(3\,2\right)$

$3$

$\mathrm{numbcomb}\left(\left[a\,a\,b\right]\right)$

$6$

$\mathrm{numbcomb}\left(\left\{a\,b\,c\right\}\right)$

$8$

$\mathrm{numbcomb}\left(\left[a\,b\,b\,c\right]\,2\right)$

$4$