Let S1=∑j=110j(j-1) Cj 10, S2=∑j=110j Cj 10 and S3=∑j=110j2 Cj 10.Statement-1 : S3=55×29Statement-2 : S1=90×28 and S2=10×28.

S1=∑j=110 j(j−1)⋅10Cj=∑j=110 j(j−1)⋅10j⋅9j−18Cj−2=90∑j=210 8Cj−2=90.28S2=∑j=110 j10Cj=∑j=110 10.9Cj−1=10 9C0+9C1+9C2+….+9C9=10.29S3=∑j=110 j2⋅10Cj=∑j=110 j2⋅10j⋅9Cj−1=10∑j=110 (j−1+1)⋅9Cj−1 =10∑j=110 (j−1)9Cj−1+∑j=110 9Cj−1=109.28+29=55.29Statement -1 is correct and Statement - 2 is False

Let S1=∑j=110j(j-1) Cj 10, S2=∑j=110j Cj 10 and S3=∑j=110j2 Cj 10.Statement-1 : S3=55×29Statement-2 : S1=90×28 and S2=10×28.

S1=∑j=110 j(j−1)⋅10Cj=∑j=110 j(j−1)⋅10j⋅9j−18Cj−2=90∑j=210 8Cj−2=90.28S2=∑j=110 j10Cj=∑j=110 10.9Cj−1=10 9C0+9C1+9C2+….+9C9=10.29S3=∑j=110 j2⋅10Cj=∑j=110 j2⋅10j⋅9Cj−1=10∑j=110 (j−1+1)⋅9Cj−1 =10∑j=110 (j−1)9Cj−1+∑j=110 9Cj−1=109.28+29=55.29Statement -1 is correct and Statement - 2 is False

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Let $S_{1} = \sum_{j = 1}^{10} j (j - 1) {}^{10}C_{j}, S_{2} = \sum_{j = 1}^{10} j {}^{10}C_{j}$ and $S_{3} = \sum_{j = 1}^{10} j^{2} {}^{10}C_{j}$ .
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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Statement-1 is false, Statement-2 is true

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for
Statement-1

Statement-1 is true, Statement-2 is false

Statement-1 is true, Statement-2 is true; Statement-2 is not correct explanation for
Statement-1

answer is C.

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Detailed Solution

$S_{1} = \sum_{j = 1}^{10} j (j - 1) \cdot^{10} C_{j} = \sum_{j = 1}^{10} j (j - 1) \cdot \frac{10}{j} \cdot {\frac{9}{j - 1}}^{8} C_{j - 2}$

$= 90 \sum_{j = 2}^{10}^{8} C_{j - 2} = {90.2}^{8}$

$S_{2} = \sum_{j = 1}^{10} j^{10} C_{j} = \sum_{j = 1}^{10} 10 .^{9} C_{j - 1}$

$= 10 [^{9} C_{0} +^{9} C_{1} +^{9} C_{2} + \dots . +^{9} C_{9}] = {10.2}^{9}$

$\begin{array}{rcl} S_{3} & = & \sum_{j = 1}^{10} j^{2} \cdot^{10} C_{j} = \sum_{j = 1}^{10} j^{2} \cdot \frac{10}{j} \cdot^{9} C_{j - 1} = 10 \sum_{j = 1}^{10} (j - 1 + 1) \cdot^{9} C_{j - 1} \\ = & 10 (\sum_{j = 1}^{10} (j - 1)^{9} C_{j - 1} + \sum_{j = 1}^{10}^{9} C_{j - 1}) = 10 ({9.2}^{8} + 2^{9}) = {55.2}^{9} \end{array}$