Question

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

• A. ${}^{501}{C}_{101}(5{)}^{399}$
• B. ${}^{501}{C}_{101}(5{)}^{400}$
• C. ${}^{501}{C}_{100}(5{)}^{400}$
• D. ${}^{500}{C}_{101}(5{)}^{399}$

Answer 1

Answer: (A)

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