We have $n$ kinds of objects, and we want to determine the number of ways in which a $k$-tuple of objects can be selected. We consider variants: we may be interested in selecting ordered or unordered $k$-tuples,
\begin{array}{c|c|c|} & \text{Only 1 object of each kind} & \text{arbitrarily many objects of each kind} \\ \hline \text{Ordered k-tuples} & \frac{n!}{(n-k)!} & k^{n}\\ \hline \text{Unordered k-tuples} & \binom{n}{k} & \binom{n+k-1}{k} \\ \hline \end{array}
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