The four basic combinatoric formulas?

We have the following cases for the number of subsets of size $k$ chosen from a set of $n$ distinct elements:

• replacement and ordered, "permutation with repetition" $$n^k$$

• no replacement and ordered, "k-permutations of n" $$\frac{n!}{(n-k)!}$$

• no replacement and unordered, "combinations" $$\binom{n}k$$

• replacement and unordered, "combination with repetitions" $$\binom{n+k-1}k$$

When order matters and repetition is allowed, you get all the functions from $\{1,2, \cdots, k\}$ to your set $S$ of $n$ elements. Every function corresponds to a unique choice, and every choice allows you to construct a function by setting $f(r)$ to be the $r^{th}$ element chosen. The formula for these is simpler, just being $n^k,$ because we have a full $n$ independent choices to make, $k$ times in a row.

Perhaps the most complicated of the four is when repetition is allowed but order does not matter. Those are called multisets. There is an explicit formula for $n$ multichoose $k$ given by $\left( {n \choose k}\right) = {n+k-1 \choose k}.$