Question

If n is a positive integer and r is a non-negative integer, prove that the coefficients of x^r and x^n−r in the expansion of (1 + x)ⁿ are equal

Answer 1

Given (1 + x)ⁿ.

General term t_r+1 = ⁿC_r x^n–r . a^r

∴ The general term in the expansion of (1 + x)ⁿ is

T^r+1 = ⁿC_r . (1)^{n– r} . x^r

T^r+1 = ⁿC_r . x^r   ......(1)

∴ Coefficient of x^r is ⁿC_r,

Put r = n – r in (1)

T_n–r+1  = ⁿC_n–r . x^n–r

∴ The coefficient of x^n-r is ⁿC_n-r   ......(2)

We know ⁿC_r = ⁿC_n-r

∴ The coefficient of x^r and coefficient of x^n-r are equal, (by (1) and  (2))