Binomial Coefficient with Self

Theorem

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

Proof
From the definition of binomial coefficients:


 * $\displaystyle \binom n n = \dfrac {n!} {n! \ \left({n - n}\right)!} = \dfrac {n!} {n! \ 0!}$

the result following directly from the definition of the factorial, where $0! = 1$.

Also see

 * Particular Values of Binomial Coefficients for other similar results.