Question

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

• A. ${}^{51}{C}_5$
• B. ${}^9{C}_5$
• C. ${}^{31}{C}_6{-}^{21}{C}_6$
• D. ${}^{30}{C}_5{-}^{20}{C}_5$

Answer 1

The correct option is C ${}^{31}{C}_6{-}^{21}{C}_6$

$(1+x{)}^{21}+(1+x{)}^{22}+...+(1+x{)}^{30}$
$=(1+x{)}^{21}\left[\frac{(1+x{)}^{10}-1}{(1+x)-1}\right]=\frac{1}{x}\left[\right(1+x{)}^{31}-(1+x{)}^{21}]$
$\Rightarrow$ Coefficient of $x^5$ in the given expression is equal to coefficient of $x^6$ in $\left[\right(1+x{)}^{31}-(1+x{)}^{21}]$, which is given by ${}^{31}{C}_6{-}^{21}{C}_6$