Question

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

Answer 1

$ \displaystyle\sum_{ K =1}^{10} K ^2\left({ }^{10} C _{ K }\right)^2$
$ \displaystyle\sum_{ K =1}^{10}\left( K \cdot{ }^{10} C _{ K }\right)^2=\displaystyle\sum_{ K =1}^{10}\left(10 \cdot{ }^9 C _{ K -1}\right)^2$
$ =100 \displaystyle\sum_{ K =1}^{10}{ }^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 $L =221$