Question:
The coefficient of $x^{5}$ in the expansion of $(1+x)^{21}+(1+x)^{22}+\ldots+(1+x)^{30}$
(a) 51C5
(b) 9C5
(c) 31C6 − 21C6
(d) 30C5 + 20C5
Solution:
(c) ${ }^{31} C_{6}-{ }^{21} C_{6}$
We have $(1+x)^{21}+(1+x)^{22}+\ldots(1+x)^{30}$
$=(1+x)^{21}\left[\frac{(1+x)^{10}-1}{(1+x)-1}\right]$
$=\frac{1}{x}\left[(1+x)^{31}-(1+x)^{21}\right]$
Coefficient of $x^{5}$ in the given expansion $=$ Coefficient of $x^{5}$ in $\frac{1}{x}\left[(1+x)^{31}-(1+x)^{21}\right]$
$=$ Coefficient of $x^{6}$ in $\left[(1+x)^{31}-(1+x)^{21}\right]$
$={ }^{31} C_{6}-{ }^{21} C_{6}$
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