Binomial Coefficient with Self minus One/Proof 2

Proof
From Cardinality of Set of Subsets, $\dbinom n {n - 1}$ is the number of combination of things taken $n - 1$ at a time.

Choosing $n - 1$ things from $n$ is the same thing as choosing which $1$ of the elements to be left out.

There are $n$ different choices for that $1$ element.

Therefore there are $n$ ways to choose $n - 1$ things from $n$.