Coefficient of $x^{4}$ in $(1 + x + x^{2} + x^{3})^{11}$ is

55

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605

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990

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1120

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Solution

The correct option is C $990$
$(1 + x + x^{2} + x^{3})^{11}$
$= (1 + x^{2})^{11} (1 + x)^{11}$
$= (1 + 11 x^{2} +^{11} C_{2} x^{4} + . . . x^{22}) (1 + 11 x +^{11} C_{2} x^{2} + . . . x^{11})$
Hence coefficient of $x^{4}$ will be
$1 \times^{11} C_{4} + 11 \times^{11} C_{2} +^{11} C_{2} \times 1$
$=^{11} C_{4} + 12 \times^{11} C_{2}$
$= 330 + 660$
$= 990$

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