### Instruction: Compute \[\frac{(N-1)!(N)}{(N+1)!}.\].### Response: I notice that there are factorials in the numerator and the denominator, so I wonder if I can simplify them somehow.
I remember that the definition of a factorial is the product of all positive integers from 1 to that number.
For example, 5! = 5 \times 4 \times 3 \times 2 \times 1.
I see that the numerator has (N-1)! times N, which is the same as N!.
So I can rewrite the fraction as \[\frac{N!}{(N+1)!}.\]
Now I notice that the denominator has (N+1)! which is the same as (N+1) times N!.
So I can rewrite the fraction again as \[\frac{N!}{(N+1) \times N!}.\]
Now I see that the N! in the numerator and the denominator cancel out, leaving me with \[\frac{1}{(N+1)}.\]