Coefficient of $x^{10}$ $-$ Coefficient of $x^{15}$ in ${(1 + x + x^{2} + x^{3})}^{4} {(1 - x)}^{8}$ is

36

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52

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20

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16

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Solution

The correct option is B $20$
${(1 + x + x^{3} + x^{4})}^{4} {(1 - x)}^{8}$

$= {(1 + x + x^{3} + x^{4})}^{4} {(1 - x)}^{4} {(1 - x)}^{4}$

$= (1 - 4 x^{4} + 6 x^{8} - 4 x^{12} + x^{16}) (1 - 4 x + 6 x^{2} - 4 x^{3} + x^{4})$

$∴$ Coefficient of $x^{10} -$ Coefficient of $x^{15}$

$= 6 \times 6 - 16 = 20$

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