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Multinomial Expansion

The coefficie...

Question

The coefficient of $x^{28}$ in the expansion of $(1 + x^{3} - x^{6})^{30}$ is

A

1

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B

0

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C

^{30} C_{6}

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D

^{30} C_{3}

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Solution

The correct option is B $0$
$(1 + x^{3} - x^{6})^{30} = \sum k_{1} + k_{2} + k_{3} = 30 \frac{30!}{k_{1}! k_{2}! k_{3}!} (1)^{k_{1}} (x^{3})^{k_{2}} (- x^{6})^{k_{3}} = \sum k_{1} + k_{2} + k_{3} = 30 \frac{30!}{k_{1}! k_{2}! k_{3}!} (- 1)^{k_{3}} x^{3 k_{2} + 6 k_{3}}$

Now, for coefficient of $x^{28}$
$3 k_{2} + 6 k_{3} = 28 \Rightarrow k_{2} + 2 k_{3} = \frac{28}{3}$
As $k_{2}, k_{3} \in W$ , so it is not possible
Therefore, coefficient of $x^{28}$ is $0$ .

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Multinomial Expansion

Standard XII Mathematics

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