Question

In the expansion of $(2x+y+3z{)}^{10}$ coefficient of $x^2{y}^5{z}^3$ is

• A. $\frac{10!}{2!3!5!}$
• B. $\frac{10!}{2!3!5!}{2}^2{3}^3$
• C. $\frac{10!}{2!3!5!}{2}^3{3}^2$
• D. None of these

Answer 1

The correct option is B $\frac{10!}{2!3!5!}{2}^2{3}^3$

${\left(\right(2x+y)+3z)}^{10}$

$coefficientof{x}^2{y}^5{z}^3$

First expand ${\left(\right(2x+y)+3z)}^{10}$

$=....+{}^{10}{C}_3{\left(3z\right)}^3{(2x+y)}^7{+}^{10}{C}_3(..{.}^7{C}_2{\left(2x\right)}^2{\left(y\right)}^5+..){\left(3z\right)}^3$

$=....+{}^{10}{C}_3{\left(3z\right)}^3{(}^7{C}_2{\left(2x\right)}^2{\left(y\right)}^5)$

${\therefore }^{10}{C}_3{\times }^7{C}_2\times 3^2\times 2^2=\frac{10!}{7!3!}\times \frac{7!}{2!5!}\times 3^2\times 2^2$

$=\frac{10!}{2!3!5!}\times 3^2\times 2^2$