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) 31C621C6

(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.