Binomial Coefficient with Self minus One

Theorem

 * $\displaystyle \forall n \in \N_{>0}: \binom n {n - 1} = n$

Proof
The case where $n = 1$ can be taken separately.

From Binomial Coefficient with Zero:
 * $\displaystyle \binom 1 0 = 1$

demonstrating that the result holds for $n = 1$.

Let $n \in \N: n > 1$.

From the definition of binomial coefficients:


 * $\displaystyle \binom n {n - 1} = \dfrac {n!} {\left({n - 1}\right)! \left({n - \left({n - 1}\right)}\right)!} = \dfrac {n!} {\left({n - 1}\right)! \ 1!}$

the result following directly from the definition of the factorial.

Also see

 * Particular Values of Binomial Coefficients for other similar results.