$(1 + x)^{21} + (1 + x)^{22} + . . . + (1 + x)^{30}$ coefficient of $x^{5}$

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Solution

$(1 + x)^{21} + (1 + x)^{22} + \dots + (1 + x)^{30} = (1 + x)^{21} \frac{(1 + x)^{10} - 1}{x}$
co efficient of $x^{5}$ in $\frac{1}{x} ((1 + x)^{31} - (1 + x)^{21})$ will be equal to coefficient of $x^{6}$ in $(1 + x)^{31} - (1 + x)^{21}$
So the coefficient is $^{31} C_{6} -^{21} C_{6}$

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