# binomial coefficient

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

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

and call such numbers binomial coefficients.

## 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.

• $n\choose r$ constitute the coefficients when expanding the binomial $(x+y)^{n}$ – hence the name binomial coefficients. See Binomial Theorem.

• $n\choose r$ is the number of ways to choose $r$ objects from a set with $n$ elements.

• On the context of computer science, it also helps to see ${n\choose r}$ as the number of strings consisting of ones and zeros with $r$ ones and $n-r$ zeros. This equivalency comes from the fact that if $S$ be a finite set with $n$ elements, ${n\choose r}$ is the number of distinct subsets of $S$ with $r$ elements. For each subset $T$ of $S$, consider the function

 $X_{T}\colon S\rightarrow\{0,1\}$

where $X_{T}(x)=1$ whenever $x\in T$ and $0$ otherwise (so $X_{T}$ is the characteristic function for $T$). For each $T\in\mathcal{P}(S)$, $X_{T}$ can be used to produce a unique bit string of length $n$ with exactly $r$ ones.

## Notes

The ${n\choose r}$ notation was first introduced by von Ettinghausen [1] 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