Question

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

Answer 1

Coefficient of $a^m$ in the expansion of $(1+a{)}^{m+n}={(m+n)}_{C_m}$

Coefficient of $a^n$ in the expansion of $(1+a{)}^{m+n}={(m+n)}_{C_n}$

$={m+n}_{C_{(m+n)-n}}={m+n}_{C_m}$ $[\because n_{C_r}=n_{C_{n-r}}]$

which shows that coefficients of $a^m$ and $a^n$ are equal.