Embibe Experts Solutions for Chapter: Binomial Theorem, Exercise 3: Exercise-3

Let $S_1=\sum _{j=1}^{10}j{\left(j-1\right)}^{10}{C}_1, S_2=\sum _{j=1}^{10}{j}^{10}{C}_j$ and $S_3=\sum _{j=1}^{10}{j}^{2}10C_i$

Statement-$1$: $S_3=55\times 2^9$.

Statement-$2$: $S_1=90\times 2^8$ and $S_2=10\times 2^8$.

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The value of$\left({}^{21}C_1-{}^{10}C_1\right)+\left({}^{21}C_2-{}^{10}C_2\right)+\left({}^{21}C_3-{}^{10}C_3\right)+\left({}^{21}C_4-{}^{10}C_4\right)+\dots \dots ..+\left({}^{21}C_{10}-{}^{10}C_{10}\right)$  is