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Question: Answered & Verified by Expert
In the expansion of $\left(1+x+x^3+x^4\right)^{10}$, the coefficient of $x^4$ is
MathematicsBinomial TheoremJEE Main
Options:
  • A ${ }^{40} \mathrm{C}_4$
  • B ${ }^{10} C_4$
  • C $210$
  • D $310$
Solution:
1320 Upvotes Verified Answer
The correct answer is: $310$
$\begin{aligned} & \left(1+x+x^3+x^4\right)^{10}=(1+x)^{10}\left(1+x^3\right)^{10} \\ & =\left(1+{ }^{10} C_1 \cdot x+{ }^{10} C_2 \cdot x^2+\ldots . .\right)\left(1+{ }^{10} C_1 \cdot x^3+{ }^{10} C_2 \cdot x^6+\ldots . .\right) \\ & \therefore \text { Coefficient of } x^4={ }^{10} C_1 \cdot{ }^{10} C_1+{ }^{10} C_4=310 .\end{aligned}$

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