Counting and combinatorics

The four counting cases

Classify by order and repetition, and the formula follows without guesswork.

\frac{n!}{(n-k)!}, \quad \binom{n}{k}, \quad n^k, \quad \binom{n+k-1}{k}

Where

$\frac{n!}{(n-k)!}$: Ordered, no repetition.
$\binom{n}{k}$: Unordered, no repetition.
$n^k$: Ordered, with repetition.
$\binom{n+k-1}{k}$: Unordered with repetition — stars and bars.

Assumptions

With repeated identical items, divide by the factorial of each repeat count.

Sanity check. In a probability, count numerator and denominator under the same convention or the $k!$ will not cancel.

Where this is taught

Counting: the four cases, and how to tell them apart · COMB · Basic counting