# binomial coefficient

For integers $n\geq r\geq 0$ we define

 ${n\choose r}=\frac{n!}{(n-r)!r!}$

## Properties.

1. 1.

$n\choose r$ is an integer (proof. (http://planetmath.org/NchooseRIsAnInteger)).

2. 2.

${n\choose r}={n\choose n-r}$.

3. 3.

${n\choose r-1}+{n\choose r}={n+1\choose r}$ (Pascal’s rule).

4. 4.

${n\choose 0}=1={n\choose n}$ for all $n$.

5. 5.

${n\choose 0}+{n\choose 1}+{n\choose 2}+\cdots+{n\choose n}=2^{n}$.

6. 6.

${n\choose 0}-{n\choose 1}+{n\choose 2}-\cdots+(-1)^{n}{n\choose n}=0$ for $n>0$.

7. 7.

$\sum_{t=k}^{n}{t\choose k}={n+1\choose k+1}$.

Properties 5 and 6 are the binomial theorem  applied to $(1+1)^{n}$ and $(1-1)^{n}$, respectively, although they also have purely combinatorial meaning.

## Motivation

Suppose $n\geq r$ are integers. The below list shows some examples where the binomial coefficients appear.

## Notes

The ${n\choose r}$ notation was first introduced by von Ettinghausen  in 1826, altough these numbers have been used long before that. See this page (http://planetmath.org/PascalsTriangle) for some notes on their history. Although the standard mathematical notation for the binomial coefficients is $n\choose r$, there are also several variants. Especially in high school environments one encounters also ${C}(n,r)$ or ${C}^{n}_{r}$ for ${n\choose r}$.

Remark. It is sometimes convenient to set ${n\choose r}:=0$ when $r>n$. For example, property 7 above can be restated: $\sum_{t=1}^{n}{t\choose k}={n+1\choose k+1}$. It can be shown that ${n\choose r}$ is elementary recursive.

## References

• 1 N. Higham, Handbook of writing for the mathematical sciences, Society for Industrial and Applied Mathematics, 1998.
 Title binomial coefficient Canonical name BinomialCoefficient Date of creation 2013-03-22 11:47:25 Last modified on 2013-03-22 11:47:25 Owner matte (1858) Last modified by matte (1858) Numerical id 32 Author matte (1858) Entry type Definition Classification msc 11B65 Classification msc 05A10 Classification msc 19D55 Classification msc 19K33 Classification msc 19D10 Synonym combinations  Synonym choose Related topic PascalsRule Related topic BinomialTheorem Related topic BernoulliDistribution2 Related topic MultinomialTheorem Related topic ProofOfLucassTheorem2 Related topic Factorial  Related topic CentralBinomialCoefficient Related topic PascalsTriangle Related topic TaylorSeriesViaDivision Related topic CombinationsWithRepeatedElements Related topic NonIsomorphicGroupsOfGivenOrder Related topic AppellSeque