The coefficient of $x^{5}$ in the expression 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 C $^{31} C_{6} -^{21} C_{6}$
T.E coefficient of $x^{5}$ in $(1 + x)^{21} + . . . . . . . + (1 + x)^{30}$
$= (1 + x)^{21} [1 + (1 + x) + (1 + x)^{2} + . . . . . . + (1 + x)^{9}]$
$= (1 + x)^{21} [\frac{(1 + x)^{10} - 1}{(1 + x) - 1}]$
$= (1 + x)^{21} [\frac{(1 + x)^{10} - 1}{x}]$
$= \frac{1}{x} [(1 + x)^{31} - (1 + x)^{21}]$
coeff. is $^{31} C_{6} -^{21} C_{6}$

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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:

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

x^{5}

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