### Instruction: Compute \[\frac{(N-1)!(N)}{(N+1)!}.\].### Response: I notice that the numerator and denominator have factorials, so I wonder if I can simplify them by expanding the factorials and cancelling common factors.
I start with the denominator.
\[(N+1)! = (N+1) \times N \times (N-1) \times (N-2) \times \cdots \times 2 \times 1.\]
I notice that the numerator has N and (N-1)! as factors, so I can write it as \[N \times (N-1) \times (N-2) \times \cdots \times 2 \times 1.\]
Now I can cancel out the common factors in the numerator and denominator.
I get \[\frac{N \times (N-1) \times (N-2) \times \cdots \times 2 \times 1}{(N+1) \times N \times (N-1) \times (N-2) \times \cdots \times 2 \times 1} = \frac{1}{N+1}.\]