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7.Binomial Theorem
hard
અભિવ્યક્તિ $(5+x)^{500}+x(5+x)^{499}+x^{2}(5+x)^{498}+\ldots . x^{500}$ $x>0$ માં $x ^{101}$ નો સહુગુણક ......... છે.
A
${ }^{501} C _{101}(5)^{399}$
B
${ }^{501} C _{101}(5)^{400}$
C
${ }^{501} C _{100}(5)^{400}$
D
${ }^{500} C _{101}(5)^{399}$
(JEE MAIN-2022)
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
$=\frac{{ }^{501} C _{101} 5^{400}}{5}={ }^{501} C _{101} 5^{399}$
Standard 11
Mathematics
