Question

Question 9 In the expansion of (1 + a)^ (m + n), prove that coefficients of a^m and a^n are equal.

Class X1 - Maths -Binomial Theorem Page 171

Answer 1

concept
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In this type of questions, first of all we have to find out The general terms , in the expansion (x + y)^n by using formula T_{r+1} = nCr.x^{n-r}y^r
and then put (n - r) equal to the required power of x of which coefficient is to be find out .

solution
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The General term in the expansion of ( 1 + a)^(m+n) is
T_{r+1} = (m+n)Cr.(a)^r
for finding coefficient of a^m , put r = m
T_{m+1} = (m+n)Cm.(a)^m
so, coefficient of a^m = (m+n)Cm--------(1)

for finding coefficient of a^n , put r = n
T_{n+1} = (m+n)Cn.(a)^n
so, coefficient of a^n = (m+n)Cn
= (m+n)C(m+n-m)
we know,
nCr = nC(n-r) use this here,

= (m+n)C(m+n-m) = (m+n)Cm -------(2)

hence, from equations (1) and (2) ,
we see that ,
coefficient of a^m = a^n
hence, proved