Definition of binomial coefficient

Question

I have this problem that I am a bit unsure about how to proceed forward with.

Problem: Show that $n{\binom{m+n}{m} = (m+1)\binom{m+n}{m+1}}$ for all integers n, m > 0.

In the solution it says that we should use the definition of binomial coefficient.

Can anyone describe or tell me how to proceed with this problem ?

Answer 1

$$\begin{align} n\binom{m+n}m &=n\frac{(m+n)!}{m!\ n!}\\ &=\frac{(m+n)!}{m!\ (n-1)!}\\ &=\color{blue}{(m+1)}\frac{(m+n)!}{\color{blue}{(m+1)}\ m!\ (n-1)!}\\ &=(m+1)\frac{(m+n)!}{(m+1)!\ (n-1)!}\\ &=(m+1)\binom{m+n}{m+1}\\ \end{align}$$