The coefficient of x5 in the expansion of (1+x)21+(1+x)22+(1+x)23 +……….. + (1+x)30 is

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

A
$51_{C_{5}}$
B
$9_{C_{5}}$
C
$31_{C_{6}} - 21_{C_{6}}$
D
$31_{C_{6}} + 20_{C_{5}}$

Solution:

$(1 + x)^{21} + (1 + x)^{22} + \dots + (1 + x)^{30}$
$= (1 + x)^{21} [\frac{(1 + x)^{10} - 1}{(1 + x) - 1}] = \frac{1}{x} [(1 + x)^{31} - (1 + x)^{21}]$
$\Rightarrow Coefficient of x^{5} in the given expression = Coefficient of x^{5} in \{\frac{1}{x} [(1 + x)^{31} - (1 + x)^{21}]\} = Coefficient of x^{6} in [(1 + x)^{31} - (1 + x)^{21}] =^{31} C_{6} -^{21} C_{6}$

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