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  4. The coefficient of x 101 in the expression (5+x)500+x(5+x)499+x2(5+x)498+ ldots . x500 x >0, is

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Q. The coefficient of $x ^{101}$ in the expression $(5+x)^{500}+x(5+x)^{499}+x^{2}(5+x)^{498}+\ldots . x^{500} x >0$, is

JEE MainJEE Main 2022Binomial Theorem

Solution:

$(5+x)^{500}+x(5+x)^{499}+x^{2}(5+x)^{498}+\ldots .+x^{500}$
$=\frac{(5+x)^{501}-x^{501}}{(5+x)-x}=\frac{(5+x)^{501}-x^{501}}{5}$
$\Rightarrow$ coefficient $x^{101}$ in given expression
$=\frac{{ }^{501} C_{101} 5^{400}}{5}={ }^{501} C_{101} 5^{399}$