Counting arrangements two ways

Write the counting formulas yourself, and test them against a computer that lists every arrangement.

The idea. ${}^nP_r = n(n - 1)\cdots(n - r + 1)$. Listing with itertools.permutations must give the same number.

Your tasks.

1. In factorial, return $n!$ with a loop. Remember that $0! = 1$.
2. In npr, return ${}^nP_r$ as a product of $r$ factors. Do not use factorial.
3. In by_listing, return how many arrangements of $r$ of the items there are, by listing them.