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Multinomial Expansion

The coefficie...

Question

The coefficient of $x^{11}$ in the expansion of $(1 + 2 x + 2 x^{2})^{6}$ is

A

144

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B

288

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C

216

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D

384

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Solution

The correct option is D $384$
Given expansion is $(1 + 2 x + 2 x^{2})^{6}$
The general term
$= \frac{6!}{k_{1}! k_{2}! k_{3}!} (1)^{k_{1}} (2 x)^{k_{2}} (2 x^{2})^{k_{3}} = \frac{6!}{k_{1}! k_{2}! k_{3}!} (2)^{k_{3} + k_{2}} (x)^{k_{2} + 2 k_{3}}$
Now,
$k_{2} + 2 k_{3} = 11 k_{1} + k_{2} + k_{3} = 6$

So,
$k_{1} = 0, k_{2} = 1, k_{3} = 5$

Therefore, the required coefficient
$= \frac{6!}{0! 1! 5!} (2)^{6} = 6 \times 64 = 384$

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Multinomial Expansion

Standard XII Mathematics

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