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Binomial Coefficients

In the expans...

Question

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

A

^{40} C_{4}

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B

^{10} C_{5}

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C

210

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D

310

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Solution

The correct option is D $310$
$(1 + x + x^{3} + x^{4})^{10} = ((1 + x) + x^{3} (1 + x))^{10}$
$= (1 + x)^{10} (1 + x^{3})^{10}$
By applying binomial expansion, we get
$= (1 +^{10} C_{1} x +^{10} C_{2} x^{2} +^{10} C_{3} x^{3} +^{10} C_{4} x^{4} . . .) (1 +^{10} C_{1} x^{3} +^{10} C_{2} x^{6} + . . .)$
Therefore, coefficient of $x^{4}$ is $^{10} C_{1}^{10} C_{1} +^{10} C_{4} = 100 + 210 = 310$

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Binomial Coefficients

Standard XII Mathematics

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