Question

Find the coefficient of $x^{-1}$ for the exp $(1+3x^2+x^4)(1+8c_1\frac{1}{x}+8c_2\frac{1}{x^2}+8c_3\frac{1}{x^3}+.....)$.

Answer 1

$(1+3x^2+x^4)(1+8C_1\frac{1}{x}+8C_2\frac{1}{x}+8C_2\frac{1}{x_2}+....)$

Co-efficient of $x^{-}1$ of the product

${}^8{C}_1+3^8{C}_3{+}^8{C}_5=\frac{8!}{(8-1)!1!}+3\times \frac{8!}{(8-3)!3!}+\frac{8!}{(8-5)!5!}=\frac{8\times 7!}{7!}+3\times \frac{8\times 7\times 6\times 5!}{5!\times 3\times 2}+\frac{8\times 7\times 6\times 5!}{3\times 2\times 1\times 5!}=8+168+56=232$