The coefficient of x 301 in 1 + x 500 + x 1 + x 499 + x 2 1 + x 498 + … . . + x 500 is:

We have,

${\left(1+x\right)}^{500}+x{\left(1+x\right)}^{499}+x^2{\left(1+x\right)}^{498}+\dots ..+x^{500}$

This is a geometric progression.

$={\left(1+x\right)}^{500}·\left\{\frac{1-{\left({\displaystyle \frac{x}{1+x}}\right)}^{501}}{1-{\displaystyle \frac{x}{1+x}}}\right\}$

$={\left(1+x\right)}^{500}\frac{\left({\left(1+x\right)}^{501}-x^{501}\right)}{{\left(1+x\right)}^{501}}·\left(1+x\right)$

$=(1+x{)}^{501}-x^{501}$

Coefficient of $x^{301}$ in $(1+x{)}^{501}-x^{501}$ is given by

${}^{501}C_{301}={}^{501}\mathrm{C}_{200}$