Consider the set of odd integers $S=\{1,3,5,7,9,11,13,15,17,19,21\}$ . Let    P(S) denote the set of subsets of S. Let$T\in P\left(S\right)$  , we define $R_T$ be the sum of elements of T and  $K=\sum _{T\in P\left(S\right)}{R}_T$ . The value of $\left[\frac{K}{{100.2}^{10}}\right]$ is ________([.] denotes GIF)

Each element in subsets appear as the number of subsets of S
1 element subsets =  $10C_0$
2 element subsets =  $10C_1$
3 element subsets =  $10C_2$
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11 element subsets =  $10C_{10}$
Number of times an element ‘i’ appears is $2^{10}$  times
 $K=(1+3+5+7+9+-----------+21)2^{10}=\left(121\right)\left(1024\right)$