Properties of Binomial Coefficients

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

Symmetry Rule

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

This is proved in Symmetry Rule for Binomial Coefficients.

Factors
For all $$r \in \R, k \in \Z$$:
 * $$k \binom r k = r \binom {r - 1} {k - 1}$$

from which:
 * $$\binom r k = \frac r k \binom {r - 1} {k - 1}$$ (if $$k \ne 0$$)

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

Also:
 * $$\binom r k = \frac r {r - k} \binom {r - 1} k$$ (if $$r \ne k$$)

These are proved in Factors of Binomial Coefficients.

Pascal's Rule
For positive integers $$n, k \,\!$$ with $$1 \leq k \leq n \,\!$$:
 * $$\binom n {k-1} + \binom n k = \binom {n+1} k$$

This is proved on the page on Pascal's Rule.

Sum of All Coefficients

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

This is proved in Sum of Binomial Coefficients for Given n.

Alternating Sum and Difference of All Coefficients

 * $$\sum_{i=0}^n \left({-1}\right)^i \binom n i = 0$$ for all $$n > 0$$.

This is proved in Alternating Sum and Difference of Binomial Coefficients for Given n.

Particular Values

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

This follows directly from the definition.

where $$T_n$$ is the $n$th triangular number.
 * $$\forall n \in \Z, n \ge 0: \binom n 2 = T_n$$

This is proved in Closed Form for Triangular Numbers: Proof using Binomial Coefficients.