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

$\ds \dbinom n k$

$=$

$\ds \dbinom {n - 1} k + \dbinom {n - 1} {k - 1}$

$\ds $

$=$

$\ds \paren {\dbinom {n - 2} {k - 1} + \dbinom {n - 2} k} + \paren {\dbinom {n - 2} {k - 2} + \dbinom {n - 2} {k - 1} }$

$\ds $

$=$

$\ds \dbinom {n - 2} {k - 2} + 2 \dbinom {n - 2} {k - 1} + \dbinom {n - 2} k$

simplifying

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 $n$.

$\blacksquare$

1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.2$ The Binomial Theorem: Problems $1.2$: $2$