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

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Solution

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

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

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

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

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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