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 write^{1}
\[\binom nk=\frac{n(n-1)\cdots(n-k+1)}{k(k-1)\cdots 2\cdot 1}=\frac{n !}{k !\cdot (n-k) !}.\]
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. ↩