nCr three ways

Work out $^{n}C_{r}$ yourself in three ways, and check that they agree with Python's math.comb.

The idea. The formula is $\dfrac{n!}{r!\,(n - r)!}$. It is quicker to multiply $r$ fractions: $\dfrac{n}{1} \times \dfrac{n - 1}{2} \times \cdots \times \dfrac{n - r + 1}{r}$. Listing the selections with itertools.combinations checks both.

Your tasks.

1. In factorial, return $n!$ with a loop. Remember $0! = 1$.
2. In choose, return $\dfrac{n!}{r!\,(n - r)!}$, using your factorial.
3. In choose_fast, multiply step by step: start with c = 1, and for k from 1 to r, set c = c * (n - r + k) // k.
4. In count_list, return how many selections of r items itertools.combinations makes from range(n).