elementary number theory - Division of Factorials

I have a partition of a positive integer $(p)$. How can I prove that the factorial of $p$ can always be divided by the product of the factorials of the parts?

As a quick example $\frac{9!}{(2!3!4!)} = 1260$ (no remainder), where $9=2+3+4$.

I can nearly see it by looking at factors, but I can't see a way to guarantee it.

Answer

The key observation is that the product of $n$ consecutive integers is divisible by $n!$. This can be proved by induction.