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7.Binomial Theorem
hard
$(1+x)^{500}+x(1+x)^{499}+x^2(1+x)^{498}+\ldots . .+x^{500}$ માં $x ^{301}$નો સહગુણક $........$ છે.
A
${ }^{501} C _{302}$
B
${ }^{500} C _{301}$
C
${ }^{500} C _{300}$
D
${ }^{501} C _{200}$
(JEE MAIN-2023)
Solution
$(1+x)^{500}+x(1+x)^{499}+x^2(1+x)^{498}+\ldots .+x^{500}$
$=(1+x)^{500} \cdot\left\{\frac{1-\left(\frac{x}{1+x}\right)^{501}}{1-\frac{x}{1+x}}\right\}$
$=(1+x)^{500} \frac{\left((1+x)^{501}-x^{501}\right)}{(1+x)^{501}} \cdot(1+x)$
$=(1+x)^{501}-x^{501}$
Coefficient of $x^{301}$ in $(1+x)^{501}-x^{501}$ is given by ${ }^{501} C _{301}={ }^{501} C _{200}$
Standard 11
Mathematics
