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Binomial Coefficients

The coefficie...

Question

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

^{n} C_{4}

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^{n} C_{4} +^{n} C_{2}

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^{n} C_{4} +^{n} C_{2} +^{n} C_{2}

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^{n} C_{4} +^{n} C_{2} +^{n} C_{1} \cdot^{n} C_{2}

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Solution

The correct option is B $^{n} C_{4} +^{n} C_{2} +^{n} C_{1} \cdot^{n} C_{2}$
$(1 + x + x^{2} + x^{3})^{n} = (1 + x)^{n} (1 + x^{2})^{n}$
coefficient of $x^{4}$ is $^{n} C_{4}^{n} C_{0} +^{n} C_{2}^{n} C_{1} +^{n} C_{0}^{n} C_{2} =^{n} C_{4} +^{n} C_{2} +^{n} C_{1}^{n} C_{2}$

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