If sum limits_k=1^10 K^2(10_C_K)^2=22000L , then L is equal to ________.

Given, sum limits_k = 1^10 k^2( ^10C_k )^2 = 2200 L => sum limits_k = 1^10 ( k . ^10C_k )^2 = 22000 L => sum limits_k = 1^10 ( k . 10 / k . ^9C_k - 1 )^2 = 22000 L => sum limits_k = 1^10 ( 10 . ^9C_k - 1 )^2 = 22000 L => 100 . sum limits_k = 1^10 ( ^9C_k - 1 )^2 = 22000 L => 100( ( ^9C_0 )^2 + ( ^9C_1 )^2 + .... + ( ^9C_9 )^2 ) \ = 22000 L => 100( ^18C_9 ) = 22000 L [Note : ( ^nC_1 )^2 + ( ^nC_2 )^2 + .... + ( ^nC_n )^2 = ^2nC_n] => 100 x 18! / 9! 9! = 22000 L => L = 221

If sum k1 10 K2 10CK2 22000L then L is equal to | null (Mathematics)

Given,

\sum\limits_{k = 1}^{10} {{k^2}{{\left( {{}^{10}{C_k}} \right)}^2} = 2200\,L}

\Rightarrow \sum\limits_{k = 1}^{10} {{{\left( {k\,.\,{}^{10}{C_k}} \right)}^2} = 22000\,L}

\Rightarrow \sum\limits_{k = 1}^{10} {{{\left( {k\,.\,{{10} \over k}\,.\,{}^9{C_{k - 1}}} \right)}^2} = 22000\,L}

\Rightarrow \sum\limits_{k = 1}^{10} {{{\left( {10\,.\,{}^9{C_{k - 1}}} \right)}^2} = 22000\,L}

\Rightarrow 100\,.\,\sum\limits_{k = 1}^{10} {{{\left( {{}^9{C_{k - 1}}} \right)}^2} = 22000\,L}

\Rightarrow 100\left( {{{\left( {{}^9{C_0}} \right)}^2} + {{\left( {{}^9{C_1}} \right)}^2}\, + \,....\, + \,{{\left( {{}^9{C_9}} \right)}^2}} \right) \\ = 22000\,L

\Rightarrow 100\left( {{}^{18}{C_9}} \right) = 22000\,L

[Note : ${\left( {{}^n{C_1}} \right)^2} + {\left( {{}^n{C_2}} \right)^2}\, + \,....\, + \,{\left( {{}^n{C_n}} \right)^2} = {}^{2n}{C_n}$ ]

\Rightarrow 100 \times {{18!} \over {9!\,9!}} = 22000\,L

\Rightarrow L = 221

Explanation

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