### 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 by canceling out some common factors.
I recall that the factorial function is defined as the product of all positive integers from 1 to n, where n is the argument.
So for example, 5! = 5 \times 4 \times 3 \times 2 \times 1.
I see that the numerator has (N-1)! and N, which means I can rewrite it as N \times (N-1)!.
This is the same as N!.
The denominator has (N+1)!, which means it is the product of all positive integers from 1 to (N+1).
But I can also write it as (N+1) \times N!, since N! is the product of all positive integers from 1 to N.
Now I have \[\frac{N!}{(N+1) \times N!}.\]
I can cancel out the N! terms in the numerator and the denominator, leaving me with \[\frac{1}{(N+1)}.\]
This is the simplest form of the expression, so I am done.