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Generalized permutations and combinations video2

Theorem 1

For any pair of positive integers n and k, {the number of k-tuples of positive integers whose sum is n} is equal to {the number of (k − 1)-element subsets of a set with n − 1 elements}, which is given by $\binom{n-1}{k-1}$

See Number of ways to put n objects in k bins, with each having at least one object

Theorem 2

For any pair of positive integers n and k, the number of k-tuples of non-negative integers whose sum is n is equal to the number of multisets of cardinality k − 1 taken from a set of size n + 1, which is given by the multiset number, $\left(\binom{n+1}{k-1}\right) = \binom{n+k-1}{k-1}$.

See Number of ways to put n objects in k bins