In the expansion of $(1 + x + x^{3} + x^{4})^{10}$ , the coefficient of $x^{4}$ is

^{40} C_{4}

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^{10} C_{4}

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210

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310

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Solution

The correct option is D $310$
$(1 + x + x^{3} + x^{4})^{10}$
$= (1 (1 + x) + x^{3} (1 + x))^{10}$
$= [(1 + x^{3}) (1 + x)]^{10}$
$= (1 + x^{3})^{10} (1 + x)^{10}$
$= (1 + 10 x^{3} + 45 x^{6} . . .) (1 + 10 x + 45 x^{2} 120 x^{3} + 210 x^{4} + . . .)$
Hence, separating the term with $x^{4}$ gives us
$(1 \times 210) x^{4} + (10 \times 10)) (x^{3} \times x)$
$= 210 x^{4} + 100 x^{4}$
$= 310 x^{4}$
Hence, the required coefficient is $310$ .

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