Question

The coefficient of $x^3{y}^4{z}^5$ in the expansion of ${(xy+yz+xz)}^6$ is

• A. $70$
• B. $60$
• C. $50$
• D. none of these

Answer 1

The correct option is B

$60$

Explanation for the correct option:

Given the expression ${(xy+yz+xz)}^6$

when we expand general term we get
${(xy+yz+xz)}^6=.......+\frac{6!}{a!b!c!}{\left(xy\right)}^a{\left(yz\right)}^b{\left(zx\right)}^c$

${(xy+yz+xz)}^6=.......+\frac{6!}{a!b!c!}{x}^{a+c}{y}^{a+b}{z}^{b+c}$

On comparing the powers $x^3{y}^4{z}^5$ we get,

$$\begin{gathered} a+c=3 \\ b+c=5 \\ a+b=4 \end{gathered}$$ On Solving these equations, we get

$\Rightarrow a=1,b=3,c=2$

Hence coefficient of $x^3{y}^4{z}^5=\frac{6!}{1!3!2!}=60$

Hence option B is correct answer