7.Binomial Theorem
medium

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

$^{30}{C_5}{ + ^{20}}{C_5}$

Solution

(c) ${(1 + x)^{21}} + {(1 + x)^{22}} + …. + {(1 + x)^{30}}$ 

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

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

$\therefore$ Coefficient of $x^5$ in the given expression 

= Coefficient of $x^5$ in $\left\{ {\frac{1}{x}[{{(1 + x)}^{31}} – {{(1 + x)}^{21}}]} \right\}$ 

= Coefficient of $x^6$ in $[{(1 + x)^{31}} – {(1 + x)^{21}}]$ = ${}^{31}{C_6} – {}^{21}{C_6}$.

Standard 11
Mathematics

Similar Questions

Start a Free Trial Now

Confusing about what to choose? Our team will schedule a demo shortly.