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Let ${s_1} = \mathop \sum \limits_{j = 1}^{10} j\left( {j - 1} \right)\left( {\begin{array}{*{20}{c}}{10}\\j\end{array}} \right)\;,$$\;{s_2} = \mathop \sum \limits_{j = 1}^{10} j\;\left( {\begin{array}{*{20}{c}}{10}\\j\end{array}} \right)\;and,$${s_3} = \mathop \sum \limits_{j = 1}^{10} {j^2}\left( {\begin{array}{*{20}{c}}{10}\\j\end{array}} \right)\;,\;$
Statement $-1$:${s_3} = 55 \times {2^9}$
Statement $-2$: ${s_1} = 90 \times {2^8}\;$ and ${s_2} = 10 \times {2^8}$
Statement $-1$ is true, Statement$-2$ is true; Statement $-2$ is not a correct explanation for Statement $-1$
Statement $-1$ is true, Statement$-2$ is true; Statement $-2$ is a correct explanation for Statement $-1$
Statement $-1$ is false, Statement$-2$ is true
Statement $-1$ is true, Statement$-2$ is false
Solution
$S_{1} =\sum j(j-1)^{10} C_{j}$
$=\sum j(j-1) \cdot \frac{10(10-1)}{(j-1)} \cdot^{8} C_{j-2}$
$ = 9 \times 10\sum\limits_{j = 2}^{10} {^8{C_{j – 2}}} = 90 \times {2^8}$
${S_2} = \sum\limits_{j = 1}^{10} j { \cdot ^{10}}{C_j} = 10\sum\limits_{j = 1}^{10} {^9{C_{j – 1}}} = 10 \times {2^9}$
${S_3} = \sum\limits_{j = 1}^{10} {{j^2}} { \cdot ^{10}}{C_j} = \sum\limits_{j = 1}^{10} {(j(j – 1) + j)} { \cdot ^{10}}{C_j}$
$ = \sum\limits_{j = 1}^{10} {j{{(j – 1)}^{10}}{C_j}} + \sum\limits_{j = 1}^{10} {j{.^{10}}{C_j}} $
$=90 \cdot 2^{8}+10 \cdot 2^{9}=(45+10) 2^{9}=55 \cdot 2^{9}$
Then statement $-1$ is true and statement $-2$ is false
