The coefficient of $x^{4}$ in the expansion of $(1 + x + x^{2} + x^{3})^{11}$ is:

991

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990

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989

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Solution

The correct option is B $990$
We have,
$\begin{matrix} {(1 + x + x^{2} + x^{3})}^{11} = {(1 + x + x^{2} (1 + x))}^{11} \Rightarrow {(1 + x)}^{11} {(1 + x^{2})}^{11} (c o e f f i c i e n t o f 1 i n {(1 + x)}^{11}) \cdot (c o e f f i c i e n t o f x^{4} i n {(1 + x^{2})}^{11}) \cdot (c o e f f i c i e n t o f x^{2} i n {(1 + x)}^{11}) + (c o e f f i c i e n t o f x^{2} i n {(1 + x^{2})}^{11}) + (c o e f f i c i e n t o f x^{4} i n {(1 + x)}^{11}) \cdot (c o e f f i c i e n t o f 1 i n {(1 + x^{2})}^{11}) \Rightarrow 1 \times^{11} C_{2} +^{11} C_{2} \times^{11} C_{1} +^{11} C_{4} \times 1 \Rightarrow \frac{11!}{a! 2!} + \frac{11!}{a! 2!} \times 11 + \frac{11!}{7! 4!} \Rightarrow 55 + 605 + 380 = 990 \end{matrix}$

Hence, this is the answer.

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