Question

If 0 ≤ r ≤ n, then the coefficient of x^r in the expansion of P = 1 + (1 + x) + (1 + x)² + (1 + x)³ + ..... + (1 + x)ⁿ is 

(a) ⁿC_r 

(b) ⁿC_{r + 1} 

(c) ^{n + 1}C_r 

(d) ^{n + 1}C_{r + 1}

Answer 1

Answer : (d)  ^{n + 1}C_{r + 1} 

Given, P = 1 + (1 + x) + (1 + x)² + (1 + x)³ + ..... + (1 + x)ⁿ 

P is the sum of a G.P of (n + 1) terms with first term = 1, common ratio = (1 + x).

Thus, the coefficient of x^r in this expansion is ^{n + 1}C_{r + 1} or ^{n + 1}C_{n – r}.