Binomial Coefficient with Self minus One/Proof 1

Theorem

$\forall n \in \N_{>0}: \dbinom n {n - 1} = n$

Proof

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

From Binomial Coefficient with Zero:

$\dbinom 1 0 = 1$

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

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

From the definition of binomial coefficients:

$\dbinom n {n - 1} = \dfrac {n!} {\paren {n - 1}! \paren {n - \paren {n - 1} }!} = \dfrac {n!} {\paren {n - 1}! \ 1!}$

the result following directly from the definition of the factorial.

$\blacksquare$