Coefficient of $x^{6}$ in ${(1 + x + x^{2})}^{6}$ is-

141

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90

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50

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140

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Solution

The correct option is D $141$
Given expression is $(1 + x + x^{2})^{6}$
General term $= \frac{6!}{a! b! c!} x^{b + 2 c}$
$b + 2 c = 6$ and $a + b + c = 6$
Thus possible values of $a, b, c$ are,
(1) $a = 3, b = 0, c = 3$
(2) $a = 2, b = 2, c = 2$
(3) $a = 1, b = 4, c = 1$
(4) $a = 0, b = 6, c = 0$
$∴$ coefficient of $x^{6}$ is
$= (\frac{6!}{0! 3! 3!} + \frac{6!}{2! 2! 2!} + \frac{6!}{4! 1! 1!} + \frac{6!}{6!}) = 141$

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