Inductive base case.

Let n=1. Then you have for LHS

And RHS

(1+1)! - 1 = 2! - 1 = 2-1 = 1. So hold for n=1.

Assume it holds for n, i.e

Try to prove it works for n+1

So for n+1 we want to get .

.

For the terms involving n take out a common factor of (n+1)! to get...

. which is what we wanted. Hence by our inductive hypothesis the equality holds.