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Mathematics Question on Binomial theorem

If
∑K=110K2(10CK)2=22000L,\begin{array}{l}\sum_{K=1}^{10}K^2\left(^{10}C_K\right)^2 = 22000L,\end{array}then L is equal to _____.

Answer

∑K=110K2(10CK)2=12 10C12+22 10C22+…+102 10C10\begin{array}{l}\sum_{K=1}^{10}K^2\left(^{10}C_K\right)^2 =1^2\ ^{10}C_1^2 + 2^2\ ^{10}C_2^2 + … + 10^2\ ^{10}C_{10}\end{array}
Let, (1+x)10=10C0+10C1x+10C2x2+….+10C10x10\begin{array}{l}\left(1 + x\right)^{10} = ^{10}C_0 + ^{10}C_1 x + ^{10}C_2 x^2 + ….+ ^{10}C_{10} x^{10} \end{array}
⇒10(1+x)9=10C1+2⋅10C2x+…+10⋅10C10x9…(1)\begin{array}{l}\Rightarrow 10\left(1 + x\right)^9 = ^{10}C_1 + 2\cdot ^{10}C_2 x +… + 10\cdot ^{10}C_{10} x^9 …\left(1\right)\end{array}
Similarly, 10(x+1)9=10⋅10C0x9+9⋅10C1x8+…+1⋅10C9\begin{array}{l}10\left(x + 1\right)^9 = 10\cdot ^{10}C_0 x^9 + 9\cdot ^{10}C_1 x^8 + … + 1\cdot ^{10}C_9\end{array}
100(1+x)18has required term with coefficient of x9\begin{array}{l}100\left(1+ x\right)^{18}\text{has required term with coefficient of} ~x^9 \end{array}
i.e.18C9100=22000L ⇒L=221\begin{array}{l}i.e. ^{18}C_9 100 = 22000 L\\\ \Rightarrow L = 221\end{array}