What does $(2n+2)!$ mean?

Question

I don't understand why $(2n+2)!$ is equal to $(2n+2)(2n+1)(2n)!$ even though I think I understand what a factorial number is (7·6·5·4·3·2·1 = 7!). Any hints?

Answer 1

Using an example of $n=3$, we have $2n+2=8, (2n+2)!=8!=8\cdot 7 \cdot 6!=(2n+2)(2n+1)(2n)!$ You are just sorting out the first two terms of the factorial.

Answer 2

$(2n+2)!=(2n+2)(2n+1)(2n)(2n-1)(2n-2)...3·2·1$ $(2n)!=2n(2n-1)(2n-2)...3·2·1$

So $(2n+2)(2n+1)(2n)!=(2n+2)(2n+1)2n(2n-1)(2n-2)...3·2·1=(2n+2)!$

Answer 3

$(2n+2)!=(2n+2)(2n+1)(2n)(2n-1)\cdots3\times 2\times 1=((2n+2)(2n+1))((2n)(2n-1)\cdots3\times 2\times 1)=(2n+2)(2n+1)(2n)!$