The number of different ways to choose $k$ out of $n$ unique items The number of different ways to choose $k$ out of $n$ unique items:


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*Without repetition and without order-significance: $\binom{n}{k}=\frac{n!}{k!(n-k)!}$

*Without repetition and with order-significance: $\binom{n}{k}\times k!=\frac{n!}{(n-k)!}$

*With repetition and with order-significance: $n^k$

*With repetition and without order-significance: $?$


Thank you!

Start by arranging all of your $n$ items in some canonical linear order. You then go through the line, picking some number of copies of each item. As you do this you say "move to the next item" $n-1$ times, and "take one of these" $k$ times. These instructions can be in any order, however, so there are $\binom{k+n-1}{k}$ ways to do this.
See also Stars and bars, in particular Theorem 2 on the linked page.
