Question

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

Answer 1

$f\left(x\right)=(1-x{)}^5(1+x+x^2+x^3{)}^4$

using summation formula of GP

$f\left(x\right)=(1-x{)}^5[\frac{1-x^4}{1-x}{]}^4$

$=(1-x)(1-x^4{)}^4$

Now first term in above expression circle have

variable of from $x^0$ & $x^1$ while the other term

will have $x^0,x^4,x^8,x^{12}$ & $x^{16}$ and the variable

of f(x) will be combination of the all variable

so to get coff of $x^{13},$ you need coff of x term I term

x coff of x form 2 term

Now coff of $x=-1,x^{12}=(-1{)}^3$ ${}^4{C}_3$

$\therefore$ coff of $x^{13}=4$