Coefficient of $x^{5}$ in $(1 + x + x^{2} + x^{3})^{10}$ is

1910

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1902

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1819

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1932

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Solution

The correct option is B $1902$
$(1 + x + x^{2} + x^{3})^{10}$

$= [(1 + x) (1 + x^{2})]^{10}$

$= (1 + x)^{10} (1 + x^{2})^{10}$

$= (^{10} C_{0} +^{10} C_{1} x +^{10} C_{2} x^{2} +^{10} C_{3} x^{3} +^{10} C_{4} x^{4} + . . . . . .) (^{10} C_{0} +^{10} C_{1} x^{2} +^{10} C_{2} x^{4} +^{10} C_{3} x^{6} +^{10} C_{4} x^{8} + . . . . . .)$

$x^{5} =^{10} C_{5}^{10} C_{0} +^{10} C_{1}^{10} C_{2} +^{10} C_{3}^{10} C_{1}$

$= 252 + 10 \times 45 + 120 \times 10$

$= 1902$

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Similar questions

\int (x + x^{2} + x^{3} + x^{4})

\frac{x}{1 - x^{2}} - \frac{x^{3}}{1 - x^{6}} + \frac{x^{5}}{1 - x^{10}} - . . . . . . . = \frac{x}{1 + x^{2}} + \frac{x^{3}}{1 + x^{6}} + \frac{x^{5}}{1 + x^{10}} + . . . . .

$x^{5} + x^{4} + x^{3} + x^{2} + x + 1$ by $x^{3} + 1$

x \neq - 1

\frac{x^{5} + x^{4} + x^{3} + x^{2}}{x^{3} + x^{2} + x + 1}

x \neq - 1

\frac{x^{5} + x^{4} + x^{3} + x^{2}}{x^{3} + x^{2} + x + 1} = 1

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