Question

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

Answer 1

In the given expression ( 1+a ) m+n we have to prove that the coefficients of a m and a n are equal.

It is known that ( r+1 ) th term in T r+1 in binomial expansion of ( a+b ) n ,

T r+1 = C n r a n−r ( b ) r

Suppose a m occurs in the ( r+1 ) th term of the expansion ( 1+a ) m+n ,

T r+1 = C m+n r ( 1 ) m+n−r ( a ) r = C m+n r ( a ) r

Comparing the indices of a in a m and in T r+1 ,

r=m

Therefore, the coefficient of a m is

C m+n m = ( m+n )! m!( m+n−m )! = ( m+n )! m!n! (1)

Suppose a n occurs in the ( k+1 ) th term of the expansion ( 1+a ) m+n ,

T k+1 = C m+n k ( 1 ) m+n−k ( a ) k = C m+n k ( a ) k

Comparing the indices of a in a n and in T k+1 ,

k=n ,

Therefore, the coefficient of a n is

C m+n n = ( m+n )! n!( m+n−n )! = ( m+n )! n!m! (2)

From equation (1) and (2), it can be observed that the coefficients of a m and a n in the expansion of ( 1+a ) m+n are equal.