Binomial Coefficient with Self minus One/Proof 1
Jump to navigation
Jump to search
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.
$\blacksquare$