# Inthe expansion of (1+x)m+n where m and n are positive integers.show that coefficient of xm and xn are equal?

(1+x)m+n

Tr+1 = nCran-rbr

Tr+1 = m+nCr(1)m+n-rxr

Comparing the indices of x in xm,

xr = xm

=> r=m

By putting the value of r, we get

Tm+1= m+nCm(1)m+n-mxm

= m+ncmxm

Coefficients= m+ncm

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

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

Comparing the indices of x in xn,

xr = xn

=> r=n

similarly, by putting the value of r we get,

Tn+1 = m+nCn(1)m+n-nxn

Coefficient= m+nCn

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

=(m+n)! / m!n!  .......(2)

From eq. (1) & (2), we can say that coefficients of xm & xn are equal.

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Dear Student!

The given expansion is (1 + x)m + n.

General term of the given expansion, tr = m + n Cr xr

Coefficient of xr in the expansion = m + n Cr

∴ Coefficient of x m in the given expansion = m + n Cm = ...(1)

Coefficient of xn in the given expansion = m + n Cn = ...(2)

From (1) and (2), we get

Coefficient of xm and xn in the given expansion are equal.

Thus, coefficient of x m in the given expansion is equal to coefficient of xn in the given expansion.

Cheers!

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