Proposition: Closed Formula For Binomial Coefficients

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

Thank you to the contributors under CC BY-SA 4.0!




  1. Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", Addison-Wesley, 1994, 2nd Edition


  1. 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.