Question

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

Answer 1

It is known that (r + 1)^th term, (T_r+1), in the binomial expansion of (a + b)ⁿ is given by

Assuming that am occurs in the (r + 1)^th term of the expansion (1 + a)^{m + n},

we obtain

Comparing the indices of a in am and in T_{r + 1},

we obtain

r = m

Therefore, the coefficient of a^m is

Assuming that an occurs in the (k + 1)^th term of the expansion (1 + a)^m+n,

we obtain

Thus, from (1) and (2), it can be observed that the coefficients of a^m and aⁿ in the expansion of (1 + a)^{m + n} are equal.