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Binomial Coefficients

The coefficie...

Question

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

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Solution

Let $E = (1 - x)^{5} (1 + x + x^{2} + x^{3})^{4}$
$= (1 - x)^{5} [(1 + x) + x^{2} (1 + x)]^{4}$
$= (1 - x)^{5} (1 + x)^{4} (1 + x^{2})^{4}$
$= (1 - x) (1 - x^{2})^{4} (1 + x^{2})^{4}$
$= (1 - x) (1 - x^{4})^{4}$
$= (1 - x) [^{4} C_{0} -^{4} C_{1} (x^{4}) +^{4} C_{2} (x^{4})^{2} -^{4} C_{3} (x^{4})^{3} +^{4} C_{4} (x^{4})^{4}]$
Coefficient of $x^{13} = (- 1) (-^{4} C_{3}) = 4$

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Binomial Coefficients

Standard XII Mathematics

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