Q. 94.3( 72 Votes )

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

Answer :

The general term T_{r+1} in the binomial expansion is given by T_{r+1} = ^{n}C_{r} a^{n-r} b^{r}

Here n= m+n , a = 1 and b= a

Putting the values in the general form

T_{r+1} = ^{m+n}C_{r} 1^{m+n-r} a^{r}

= ^{m+n}C_{r} a^{r}………….1

Now we have that the general term for the expression is,

T_{r+1} = ^{m+n}C_{r} a^{r}Now, For coefficient of a^{m}

T_{m}_{+1} = ^{m+n}Cm a^{m}Hence, for coefficient of a^{m}, value of r = m

So, the coefficient is ^{m+n}C_{m}

^{n}is

^{m+n}C

_{n}

^{m+n}C_{m} =

And also,^{m+n}C_{n = }

The coefficient of a^{m} and a^{n} are same i.e.;

**Hence proved.**

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