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

Answers (1)

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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