Combinations with repetition

Combinations with repetition count the ways of choosing $k$ items from $n$ types when items may repeat and order does not matter:

$\overline{C}_n^{\,k}=\binom{n+k-1}{k}$

Example: how many different sets of 3 balls can be chosen from 4 colours (colours may repeat)? Answer: $\binom{4+3-1}{3}=\binom{6}{3}=20$.

This is a variant of ordinary combinations — for comparison the calculator also gives the number of combinations without repetition $\binom{n}{k}$. Typical uses: distributing identical objects, selecting from an assortment, “stars and bars”.