The coefficient of x^301 in (1+x)^500+x(1+x)^499+x^2(1+x)^498++x^500 is:

Question

The coefficient of $x^{301}$ in $(1+x)^{500}+x(1+x)^{499}+x^2(1+x)^{498}+\cdots+x^{500}$ is:

Accepted Answer

The coefficient of ${x^{301}}$ in

$${(1 + x)^{500}} + x{(1 + x)^{499}} + {x^2}{(1 + x)^{498}}\, + \,...\, + \,{x^{500}}$$

$${}^{500}{C_{301}} + {}^{499}{C_{300}} + {}^{498}{C_{299}}\, + \,...\, + \,{}^{199}{C_0}$$

$$ = {}^{500}{C_{199}} + {}^{499}{C_{199}} + {}^{498}{C_{199}}\, + \,...\, + \,{}^{199}{C_{199}}$$

$= {}^{501}{C_{200}}$

Solutions · 1