Evaluate

Note that:

$(n + 2)! = (n + 2)(n + 1)(n!)$

and

$(n + 1)! = (n + 1)(n!)$

Thus,

$\displaystyle \frac{(n + 2)! - (n + 1)! - n!}{(n + 2)(n!)} =$

$\displaystyle \frac{n!\bigg[(n + 2)(n + 1) - (n + 1) - 1\bigg]}{n! (n + 2)} =$

$\displaystyle \frac{(n + 2)(n + 1) - (n + 1) - 1}{n + 2} =$

$\displaystyle \frac{n^2 + 2n}{n + 2} =$

$\displaystyle \frac{n(n + 2)}{n + 2} =$

$\boxed{n}$

Substitute n=2 into the equation

(2+2)!-(2+1)!-2! / (2+2) (2!)

= 24-6-2 / 8 = 16/8 = 2,

which is equal to n.