# Q9.    In the expansion of $(1 + a)^{m+n}$ , prove that coefficients of $a^m$ and $a^n$ are equal

P Pankaj Sanodiya

As we know that the general  $(r+1)^{th}$ term  $T_{r+1}$ in the binomial expansion of  $(a+b)^n$  is given by

$T_{r+1}=^nC_ra^{n-r}b^r$

So, the general $(r+1)^{th}$ term  $T_{r+1}$ in the binomial expansion of  $(1 + a)^{m+n}$  is given by

$T_{r+1}=^{m+n}C_r1^{m+n-r}a^r=^{m+n}C_ra^r$

Now, as we can see $a^m$ will come when $r=m$ and $a^n$ will come when $r=n$

So,

Coefficient of $a^m$ :

$K_{a^m}=^{m+n}C_m=\frac{(m+n)!}{m!n!}$

CoeficientCoefficient of $a^n$ :

$K_{a^n}=^{m+n}C_n=\frac{(m+n)!}{m!n!}$

As we can see $K_{a^m}=K_{a^n}$.

Hence it is proved that the coefficients of $a^m$ and $a^n$ are equal.

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