### Instruction: Compute \[\frac{(N-1)!(N)}{(N+1)!}.\].### Response: I see a factorial expression, so I want to simplify it by canceling out common factors.
I notice that (N+1)! is the same as (N+1)(N)(N-1)!, so I can rewrite the denominator as that.
Now I have \[\frac{(N-1)!(N)}{(N+1)(N)(N-1)!}.\]
I can cancel out the (N-1)! factors in the numerator and denominator, leaving me with \[\frac{N}{(N+1)(N)}.\]
I can also cancel out one of the N factors in the numerator and denominator, leaving me with \[\frac{1}{N+1}.\]