If Σ_{ k =1}^{10} K ^{2}left(10_{ C _{ K }}right)^{2}=22000 L , then L is equal to .....

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7.Binomial Theorem

hard

If $\sum_{ k =1}^{10} K ^{2}\left(10_{ C _{ K }}\right)^{2}=22000 L$, then $L$ is equal to $.....$

A

$222$

B

$221$

C

$223$

D

$224$

(JEE MAIN-2022)

Solution

$\sum_{ K =1}^{10} K ^{2}\left({ }^{10} C _{ K }\right)^{2}$

$\sum_{ K =1}^{10}\left( K ^{10} C _{ K }\right)^{2}=\sum_{ K =1}^{10}\left(10 \cdot{ }^{9} C _{ K -1}\right)^{2}$

$=100 \sum_{ K =1}^{9} C _{ K -1} \cdot{ }^{9} C _{10- K }$

$=100\left({ }^{18} C _{9}\right)=100\left(\frac{18 !}{9 ! 9 !}\right)$

$\Rightarrow 4862000=22000 L$

Hence $L =221$

Standard 11

Mathematics

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