In the expansion of $(1+a)^{m+n},$ prove that the coefficients of $a^{m}$ and $a^{n}$ are equal.

(N/A) The general term $(T_{r+1})$ in the binomial expansion of $(1+a)^{N}$ is given by $T_{r+1} = {}^{N}C_{r} a^{r}$.
For the expansion $(1+a)^{m+n}$,the coefficient of $a^{m}$ is obtained by setting $r=m$:
Coefficient of $a^{m} = {}^{m+n}C_{m} = \frac{(m+n)!}{m!(m+n-m)!} = \frac{(m+n)!}{m!n!}$ ........... $(1)$
Similarly,the coefficient of $a^{n}$ is obtained by setting $r=n$:
Coefficient of $a^{n} = {}^{m+n}C_{n} = \frac{(m+n)!}{n!(m+n-n)!} = \frac{(m+n)!}{n!m!}$ ........... $(2)$
Since $m!n! = n!m!$,it follows that ${}^{m+n}C_{m} = {}^{m+n}C_{n}$.
Thus,the coefficients of $a^{m}$ and $a^{n}$ are equal.