The binomial coefficient \binom nk can be for k,n\in\mathbb N, n\ge 1, n\ge k calculated using the following closed formula:
\binom nk=\frac{n^{\underline{k}}}{k!},
whereas n^{\underline{k}} denotes the falling factorial power n(n-1)\cdots(n-k+1) and k! denotes the factorial k(k-1)\cdots 2\cdot 1, i.e. we can also write1
\binom nk=\frac{n(n-1)\cdots(n-k+1)}{k(k-1)\cdots 2\cdot 1}=\frac{n !}{k !\cdot (n-k) !}.
Proofs: 1
Explanations: 1
Proofs: 2 3
Propositions: 4
Please note that from the combinatorial interpretation of the binomial coefficient (as the number of subsets of a finite set with a fixed number of elements) demonstrates that the fraction is always an integer, which is not so obvious from the formula. ↩
