Properties of Binomial Coefficients

This page gathers together some of the simpler and more common identities concerning binomial coefficients.

Symmetry Rule for Binomial Coefficients

$\displaystyle \forall n \in \Z, n > 0: \forall k \in \Z: \binom n k = \binom n {n - k}$

Factors of Binomial Coefficients

For all $r \in \R, k \in \Z$:

$\displaystyle k \binom r k = r \binom {r - 1} {k - 1}$

where $\displaystyle \binom r k$ is a binomial coefficient.

Hence:

$\displaystyle \binom r k = \frac r k \binom {r - 1} {k - 1}$ (if $k \ne 0$)

and:

$\displaystyle \frac 1 r \binom r k = \frac 1 k \binom {r - 1} {k - 1}$ (if $k \ne 0$ and $r \ne 0$)

Also, for all $r \in \R, k \in \Z$:

$\displaystyle \left ({r - k}\right) \binom r k = r \binom {r - 1} k$

from which:

$\displaystyle \binom r k = \frac r {r - k} \binom {r - 1} k$ (if $r \ne k$)

Pascal's Rule

For positive integers $n, k$ with $1 \le k \le n$:

$\displaystyle \binom n {k-1} + \binom n k = \binom {n+1} k$

This is also valid for the real number definition:

$\displaystyle \forall r \in \R, k \in \Z: \binom r {k-1} + \binom r k = \binom {r+1} k$

Sum of Binomial Coefficients for Given n

$\displaystyle \sum_{i \mathop = 0}^n \binom n i = 2^n$

Alternating Sum and Difference of Binomial Coefficients for Given n

$\displaystyle \sum_{i=0}^n \left({-1}\right)^i \binom n i = 0$ for all $n \in \Z: n > 0$

Sum of Even Index Binomial Coefficients

$\displaystyle \sum_{i \ge 0} \binom n {2 i} = 2^{n-1}$

Sum of Odd Index Binomial Coefficients

$\displaystyle \sum_{i \ge 0} \binom n {2 i + 1} = 2^{n-1}$

Rising Sum of Binomial Coefficients

$\displaystyle \sum_{j=0}^m \binom {n + j} n = \binom {n+m+1} {n+1} = \binom {n+m+1} m$

Sum of Binomial Coefficients over Upper Index

$\displaystyle \sum_{j=0}^n \binom j m = \binom {n+1} {m+1}$

Increasing Sum of Binomial Coefficients

$\displaystyle \sum_{j=0}^n j \binom n j = n 2^{n-1}$

Increasing Alternating Sum of Binomial Coefficients

$\displaystyle \sum_{j=0}^n \left({-1}\right)^{n+1} j \binom n j = 0$

Chu-Vandermonde Identity

$\displaystyle \sum_k \binom r k \binom s {n-k} = \binom {r+s} n$

Sum of Squares of Binomial Coefficients

$\displaystyle \sum_{i=0}^n \binom n i^2 = \binom {2 n} n$

Particular Values

Binomial Coefficient with Zero

$\displaystyle \forall r \in \R: \binom r 0 = 1$

Binomial Coefficient with One

$\displaystyle \forall r \in \R: \binom r 1 = r$

Binomial Coefficient with Self

$\displaystyle \forall n \in \N: \binom n n = 1$

Binomial Coefficient with Two

$\displaystyle \forall r \in \R: \binom r 2 = \frac {r \left({r - 1}\right)} 2$