Question

The coefficient of $x^3{y}^{4}z$ in the expansion of $(1+x+y-z{)}^9$ is

• A. ${2.}^9{\text{C}}_7{.}^7{\text{C}}_4$
• B. $-{2.}^9{\text{C}}_2{.}^7{\text{C}}_3$
• C. ${}^9{\text{C}}_7{.}^7{\text{C}}_4$
• D. none of these

Answer 1

The correct option is B $-{2.}^9{\text{C}}_2{.}^7{\text{C}}_3$

The required coefficient will be
$\frac{-9!}{3!4!1!1!}=\frac{-9!}{3!4!}$
Multiplying and dividing by $7!$, we get
$=-[\frac{9!}{7!}.\frac{7!}{3!4!}]$
Multiplying and dividing by $2!,$ we get
$=-2![\frac{9!}{7!2!}.\frac{7!}{3!.4!}]$
$=-2[{}^9{C}_2.{}^7{C}_3]$