Binomial Coefficient with Self minus One/Proof 2
Jump to navigation
Jump to search
Theorem
- $\forall n \in \N_{>0}: \dbinom n {n - 1} = n$
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 $n$.
$\blacksquare$