Factorial Proof Problem

Question

Suppose $m$ and $n$ are positive integers

Prove $m!n! \lt (m+n)!$

I have something along the lines of:

Since $1 \lt m+1$ and $2 \lt m+2$ etc.. then: $$n \lt m + n$$ So: $$n! \lt (m+n)!$$ I'm stumped after this step. I know that I haven't enough information to conclude my proof. Any pointers?

Answer 1

$$\frac{(m+n)!}{m!n!}=\binom{m+n}{n}>1$$

Answer 2

$$n!=n(n-1)\cdots2\cdot1<(n+m)(n-1+m)\cdots(2+m)\cdot(1+m)=\frac{(m+n)!}{m!}$$