Inthe expansion of (1+x)^{m+n} where m and n are positive integers.show that coefficient of x^{m} and x^{n} are equal?

(1+x)^{m+n}

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

T_{r+1 }= ^{m+n}C_{r}(1)^{m+n-r}x^{r}

Comparing the indices of x in x^{m},

x^{r} = x^{m}

=> r=m

By putting the value of r, we get

T_{m+1}= ^{m+n}C_{m}(1)^{m+n-m}x^{m}

= ^{m+n}c_{m}x^{m}

Coefficients= ^{m+n}c_{m}

=(m+n)! / (m+n-m)!m!

=(m+n)! / m!n! .........(1)

Comparing the indices of x in x^{n},

x^{r} = x^{n}

=> r=n

similarly, by putting the value of r we get,

T_{n+1} = ^{m+n}C_{n}(1)^{m+n-n}x^{n}

Coefficient= ^{m+n}C_{n}

= (m+n)! / (m+n-n)!n!

=(m+n)! / m!n! .......(2)

From eq. (1) & (2), we can say that coefficients of x^{m} & x^{n are equal.}

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