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  4. The coefficient of x13 in the expansion of (1 - x)5(1 + x + x2 + x3)4 is

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Q. The coefficient of $x^{13}$ in the expansion of $\left(1 - x\right)^{5}\left(1 + x + x^{2} + x^{3}\right)^{4}$ is

NTA AbhyasNTA Abhyas 2020Binomial Theorem

Solution:

Let, $E=\left(1 - x\right)^{5}\left(\left(1 + x\right) \left(1 + x^{2}\right)\right)^{4}$
$=\left(1 - x\right)\left(1 - x\right)^{4}\left(1 + x\right)^{4}\left(1 + x^{2}\right)^{4}$
$=\left(1 - x\right)\left(1 - x^{2}\right)^{4}\left(1 + x^{2}\right)^{4}$
$=\left(1 - x\right)\left(1 - x^{4}\right)^{4}$
$=\left(1 - x\right)\left(1 - 4 x^{4} + 6 x^{8} - 4 x^{12} + x^{16}\right)$
Coefficient of $x^{13}$ is $\left(- 1\right)\left(- 4\right)=4$