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

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

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^{31} C_{6} +^{21} C_{6}

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2.^{31} C_{6} + 3.^{21} C_{6}

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^{31} C_{6} + 10 \cdot^{21} C_{6}

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Solution

The correct option is A $^{31} C_{6} -^{21} C_{6}$
$(1 + x)^{21} + (1 + x)^{22} + (1 + x)^{23} + \dots + (1 + x)^{30} = \frac{(1 + x)^{21} {(1 + x)^{10} - 1}}{(1 + x) - 1} = \frac{1}{x} [(1 + x)^{31} - (1 + x)^{21}]$
coefficient of $x^{5}$ = coefficient of $x^{6}$ in $[(1 + x)^{31} - (1 + x)^{21}]$
coefficient of $x^{5}$ $=^{31} C_{6} -^{21} C_{6}$

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