The coefficient of $x^{13}$ in the expansion of $(1-x)^{5}(1+x+x^{2}+x^{3})^{4}$ is
• A
$4$
• B
$-4$
• C
$0$
• D
none of these

#### Solution

Given, $(1-x)^{5}(1+x+x^2+x^3)^{4}$
$=(1-x)^{5}((1+x)(1+x^2))^{4}$
$=(1-x)(1-x^2)^{4}(1+x^2)^{4}$
$=(1-x)(1-x^4)^{4}$
$=(1-x^{4})^{4}-x(1-x^{4})^{4}$
Hence the coefficient of $x^{13}$ will be
$=-[(-1)^{3}\:^{4}C_{3}]$
$=4$