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The coefficient of $x^{5}$ in the expansion of ${(1 + x)}^{21} + {(1 + x)}^{22} + {(1 + x)}^{23} + . . . . . . . . . . . + {(1 + x)}^{30}$ is

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51C5

9C5

31C6-21C6

31C6+20C5

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detailed solution

Correct option is C

(1+x)21+(1+x)22+…+(1+x)30=(1+x)21(1+x)10−1(1+x)−1=1x(1+x)31−(1+x)21⇒ Coefficient of x5 in the given expression = Coefficient of x5 in 1x(1+x)31−(1+x)21 = Coefficient of x6 in (1+x)31−(1+x)21 =31C6−21C6

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The coefficient of x5 in the expansion of (1+x)21+(1+x)22+(1+x)23 +........... + (1+x)30 is

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The coefficient of $x^{5}$ in the expansion of ${(1 + x)}^{21} + {(1 + x)}^{22} + {(1 + x)}^{23} + . . . . . . . . . . . + {(1 + x)}^{30}$ is