The coefficient of (x^{301}) in ((1+x)^{500}+x(1+x)^{499}+x^... | Mathematics | SolveeIt

Question

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

${ }^{500} C _{301}$

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

${ }_3{ }^{500} C_{300}$

${ }^{501} C_{302}$

Answer

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

Explanation

Solution

(1+x)500+x(1+x)499+x2(1+x)498+…+x500
=(1+x)500⋅{1−1+xx1−(1+xx)501}
=(1+x)500(1+x)501((1+x)501−x501)⋅(1+x)
=(1+x)501−x501
Coefficient of x301 in (1+x)501−x501 is given by
501C301=501C200

Mathematics Question on general and middle terms

Solution