The coefficient of x^5 in the expansion of $ (1 + x)^{21} +(1+x)^{22} +…+ (1 + x)^{30} $ is

Q: The coefficient of x⁵ in the expansion of $\large (1 + x)^{21} +(1+x)^{22} +…+ (1 + x)^{30} $ is

(A) ⁵¹C₅

(B) ⁹C₅

(C) ³¹C₆ – ²¹C₆

(D) ³⁰C₅ + ²⁰C₅

Sol: $\large (1 + x)^{21} +(1+x)^{22} +…+ (1 + x)^{30} $

$\large = (1 + x)^{21} (\frac{(1+x)^{10}-1}{(1+x)-1})$

$\large = \frac{1}{x} [(1+x)^{31} – (1+x)^{21} ] $

coefficient of x⁵ in the given expression

$\large = coefficient \; of \; x^5 in \; \frac{1}{x} [(1+x)^{31} – (1+x)^{21} ] $

$\large = coefficient \; of \; x^6 in \; [(1+x)^{31} – (1+x)^{21} ] $

$\large = 31_{C_6} – 21_{C_6} $

Hence (C) is the correct answer.