Binomial Coefficient n Choose j in terms of n-2 Choose r

Theorem
Let $n \in \Z$ such that $n \ge 4$.

Let $\dbinom n k$ denote a binomial coefficient for $k \in \Z$.

Then:


 * $\dbinom n k = \dbinom {n - 2} {k - 2} + 2 \dbinom {n - 2} {k - 1} + \dbinom {n - 2} k$

for $2 \le k \le n - 2$.

Proof
In the expression $\dbinom {n - 2} {k - 2} + 2 \dbinom {n - 2} {k - 1} + \dbinom {n - 2} k$ we note that:


 * if $k < 2$ then $\dbinom {n - 2} {k - 2}$ has a negative coefficient on the bottom


 * if $k > n - 2$ then $\dbinom {n - 2} k$ has a coefficient on the bottom that is greater than $n$.

Hence the usual comfortable range of $k$ is exceeded and so it cannot be guaranteed that the conditions are satisfied for the equation to be true.

If $n \le 3$ then $2 \le k \le n - 2$ cannot be fulfilled.

Hence the bounds on both $k$ and $n$.