Browse Questions

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

Toolbox:
• $T_{r+1}=(m+n)C_ra^r.$
• $nC_r=nC_{n-r}$
Putting $r=m$
$T_{m+1}=(m+n)C_m$-----(1)
$\therefore$ Coefficient of $a^m=(m+n)C_m$
Again putting $r=n$
$T_{n+1}=(m+n)C_na^n$
Coefficient of $a^n=(m+n)C_n=(m+n)C_m$-----(2)
From (1) and (2) coefficient of $a^m$ is equal to $a^n$
Hence proved.