Question

The coefficient of x^r(0 £ r £ (n – 1)) in the expansion of

(x + 2)^n–1 + (x + 2)^n–2 (x + 1) + (x + 2)^n–3 (x + 1)² + …. +

(x + 1)^n–1

is equal to –

• A. (2^n+r + 1) ⁿC_r
• B. (2^n–r + 1) ⁿC_r
• C. (2^n–r –1) ⁿC_r
• D. (2^n+r –1) ⁿC_r

Answer 1

Answer: (C)

(2^n–r –1) ⁿC_r

Answer 2

We have,

(x + 2)^n–1 + (x + 2)^n–2 (x + 1) + (x + 2)^n–3

(x + 1)² + ….. + (x + 1)^n–1

= (x + 2)^{n – 1}$\left\{ \frac{1–\left( \frac{x + 1}{x + 2} \right)^{n}}{1 - \frac{x + 1}{x + 2}} \right\}$

= (x + 2)ⁿ – (x + 1)ⁿ

\ Coefficient of x^r in the given expression

= Coefficient of x^r in (x + 2)ⁿ – Coefficient of x^r in (x + 1)ⁿ

= ⁿC_r 2^n–r – ⁿC_r

= (2^n–r – 1) ⁿC_r

Hence (3) is correct answer.