Question

The coefficient of $x^2{y}^3{z}^4$ in the expansion of $(ax-by+cz{)}^9$ is

• A. $1260a^2{b}^3{c}^4$
• B. $-1260a^2{b}^3{c}^4$
• C. $1220a^2{b}^3{c}^4$
• D. $-1220a^2{b}^3{c}^4$

Answer 1

The correct option is B $-1260a^2{b}^3{c}^4$

Using multi-nomial theorem we get coefficient of
$x^p{y}^m{z}^l$ in $(ax+by+cz{)}^n$
$=\frac{n!\left(a^p{b}^m{c}^l\right)}{p!m!l!}$ ...(i)
Substituting the value of $n,m,l,p$ from above question to equation (i) we get,
coefficient of $x^2{y}^3{z}^4$ in $(ax-by+cz{)}^9$
$=\frac{9!\left(a^2\right(-b{)}^3{c}^4)}{2!3!4!}$
$=-\left(140\right)\left(9\right)\left(a^2{b}^3{c}^4\right)$
$=-1260\left(a^2{b}^3{c}^4\right)$