### Instruction: Compute \[\frac{(N-1)!(N)}{(N+1)!}.\].### Response: I notice that this is a fraction of factorials, so I wonder if I can simplify it by canceling out some common factors.
I start by expanding the factorials in the numerator and denominator, using the definition of factorial: \[n!=n\times(n-1)\times(n-2)\times\cdots\times 2\times 1.\]
So I get \[\frac{(N-1)!(N)}{(N+1)!}=\frac{N\times(N-1)\times(N-2)\times\cdots\times 2\times 1}{(N+1)\times N\times(N-1)\times(N-2)\times\cdots\times 2\times 1}.\]
I see that I can cancel out the common factors of N, (N-1), (N-2), ..., 2, and 1 in the numerator and denominator, leaving only (N+1) in the denominator.