### Instruction: Compute \[\frac{(N-1)!(N)}{(N+1)!}.\].### Response: I notice that this fraction has a lot of factorials, so I wonder if I can simplify it by canceling out some common factors.
I recall that a factorial means multiplying all the positive integers up to a certain number, so for example, 5! = 5 \times 4 \times 3 \times 2 \times 1.
I see that the numerator has (N-1)! and N, so I can rewrite that as N! by using the definition of factorial: N! = N \times (N-1)!
Similarly, the denominator has (N+1)!, so I can rewrite that as (N+1) \times N!.
Now the fraction looks like \[\frac{N!}{(N+1) \times N!}.\]
I notice that N! appears in both the numerator and the denominator, so I can cancel them out.
That leaves me with \[\frac{1}{N+1}.\]
This is the simplest form of the fraction, and I can't simplify it further.