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

120

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20

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30

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60

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Solution

The correct option is D $60$
${(x y + y z + z x)}^{6} = . . . . + \frac{6!}{a! b! c!} {(x y)}^{a} {(y z)}^{b} {(z x)}^{c}$
$\Rightarrow x^{a + c} y^{a + b} z^{b + c}$
$a + c = 3, b + c = 5, a + b = 4$
$∴ b - a = 2, b + a = 4$
$b = 3, a = 1, c = 2$
Coefficient of $x^{3} y^{4} z^{5} = \frac{6!}{3! 1! 2!} = 60$

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