Question

The coefficient of $x^{20}$ in the expansion of $(1+3x+3x^2+x^3{)}^{20}$ is :

• A. ${}^{60}{C}_{40}$
• B. ${}^{30}{C}_{20}$
• C. ${}^{15}{C}_2$
• D. None of these

Answer 1

Answer: (A)

${}^{60}{C}_{40}$

$(1+3x+3x^2+x^3{)}^{20}=(1+x{)}^{60}$ ..... $\left(i\right)$

In the expansion of $(a+b{)}^n,(r+1{)}^{th}$ term is given by

$t_{r+1}={}^n{C}_r\cdot a^r\cdot b^{n-r}$ or $t_{r+1}={}^n{C}_r\cdot a^{n-r}\cdot b^r$

$\therefore {21}^{st}$ term in $\left(i\right)$ will contain the coefficient of $x^{20}$ and it is given by

$t_{21}={}^{60}{C}_{20}\cdot x^{20}\cdot 1^{40}={}^{60}{C}_{20}\cdot x^{20}$
$\therefore$ Coefficient of $x^{20}$ in $(1+x{)}^{60}$ is ${}^{60}{C}_{20}$ or ${}^{60}{C}_{40}$