Simplify the expression $(2n)!/(2n+2)!$

Note that:

$$(2n+2)! = (2n+2) \cdot (2n + 1) \cdot \underbrace{2n \cdot (2n - 1) \cdot (2n - 2) \dots \cdot 2 \cdot 1}_{=(2n)!}$$

Which means $$(2n+2)! = (2n+2) \cdot (2n+1) \cdot (2n)!$$

So when dividing $(2n+2)!$ by $(2n)!$ only those first two factors of $(2n+2)!$ remain (in this case in the denominator).

Hint: One very useful property of factorials is that

$$(N + 1)! = (N + 1) N! \implies \frac{(N + 1)!}{N!} = (N + 1)$$

Similarly, using the fact that $(N + 2)! = (N + 2)(N + 1) N!$ will help simplify the desired quotient.

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