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
$$=-[(-1)^{3}\:^{4}C_{3}]$$
$$=4$$