Coefficient of x5 inÂ 1-x231-x31-x41-x5 is

1-x31-x31-x41-x5=1-x31-x31-x1+x2+x1-x1+x1+x21-x5 = (1 + x)4(x4 + x3 + 2x2 + x + 1) = 4C1 + 4C2 + 24C3 + 4C4 = 19

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Coefficient of x5 inÂ $\frac{{(1 - x^{2})}^{3} (1 - x^{3}) (1 - x^{4})}{{(1 - x)}^{5}}$ is

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$\frac{{(1 - x^{})}^{3} {(1 - x^{})}^{3} (1 - x^{4})}{{(1 - x)}^{5}} = \frac{{(1 - x^{})}^{3} {(1 - x^{})}^{3} (1 - x^{}) (1 + x^{2} + x) (1 - x) (1 + x) (1 + x^{2})}{{(1 - x)}^{5}}$
= (1 + x)4(x⁴ + x³ + 2x² + x + 1)
= ⁴C₁ + ⁴C₂ + 2⁴C₃ + ⁴C₄ = 19

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