# 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

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### References

#### Bibliography

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

#### Footnotes

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.