If ∑ _ k = 1 ^ 10 K ^ 2 ( 10 _ C _ K ) ^ 2 = 22000L , then L is...

Given,

$\mathop{∑}_{K = 1}^{10}{K}^{2}{\left(CK 10\right)}^{2} = 22000L$

Now taking L.H.S we get,

$\mathop{∑}_{K = 1}^{10}{\left({K}^{10}{C}_{K}\right)}^{2} = \mathop{∑}_{K = 1}^{10}{\left(K \times \dfrac{10}{K} . CK - 1 9\right)}^{2}$

$= \mathop{∑}_{K = 1}^{10}{\left(10 . CK - 1 9\right)}^{2}$

$= 100\mathop{∑}_{K = 1}^{10} CK - 1 9 \cdot C10 - K 9$

Now using the rule that $\mathop{∑}_{r = 1}^{n} Cr - 1 n \cdot Cn - \left(r - 1\right) n = Cn 2n$

So using the above formula we get, $\mathop{∑}_{K = 1}^{10}{\left({K}^{10}{C}_{K}\right)}^{2} = 100\left(C9 18\right) = 100\left(\dfrac{18 !}{9 ! 9 !}\right)$

$⇒ 4862000 = 22000L$

$⇒ 22000 \times 221 = 22000L$

Now on comparing we get,

Hence $L = 221$