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

^{51} C_{5}

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^{9} C_{5}

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^{31} C_{6} -^{21} C_{6}

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^{30} C_{5} +^{20} C_{5}

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Solution

The correct option is B $^{31} C_{6} -^{21} C_{6}$
We have,
$(1 + x)^{21} + (1 + x)^{22} + . . . . . + (1 + x)^{30}$

This is in a G.P with $a = (1 + x)^{21}, r = (1 + x)$

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

$= \frac{1}{x} {(1 + x)^{31} - (1 + x)^{21}}$

Therefore, coefficient of $x^{5}$ in the given expansion

$=$ Coefficient of $x^{5}$ in $[\frac{1}{x} {(1 + x)^{31} - (1 + x)^{21}}]$

$=$ Coefficient of $x^{6}$ in ${(1 + x)^{31} - (1 + x)^{21}}$

$=^{31} C_{6} -^{21} C_{6}$ .

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Similar questions

x^{5}

(1 + x)^{21} + (1 + x)^{22} + . . . + (1 + x)^{30}

{(1 + x)}^{21} + {(1 + x)}^{22} + . . . + {(1 + x)}^{30}

x^{5}

{(1 + x)}^{21} + {(1 + x)}^{22} + \dots + {(1 + x)}^{30}

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

x^{5}

(1 + x)^{21} + (1 + x)^{22} + . . . + (1 + x)^{30}

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