Let s_1 = sum_{j=1}^{10} j(j-1) binom{10}{j}, | vedclass

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(D) We know that $\sum_{j=0}^{n} \binom{n}{j} = 2^n$.For $s_1 = \sum_{j=2}^{10} j(j-1) \binom{10}{j} = \sum_{j=2}^{10} 10 	imes 9 \binom{8}{j-2} = 90

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Statement $-1$ is true,Statement $-2$ is false

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(D) We know that $\sum_{j=0}^{n} \binom{n}{j} = 2^n$.For $s_1 = \sum_{j=2}^{10} j(j-1) \binom{10}{j} = \sum_{j=2}^{10} 10 	imes 9 \binom{8}{j-2} = 90 	imes 2^8$.For $s_2 = \sum_{j=1}^{10} j \binom{10}{j} = \sum_{j=1}^{10} 10 \binom{9}{j-1} = 10 	imes 2^9$.For $s_3 = \sum_{j=1}^{10} j^2 \binom{10}{j} = \sum_{j=1}^{10} (j(j-1) + j) \binom{10}{j} = s_1 + s_2$.$s_3 = 90 	imes 2^8 + 10 	imes 2^9 = 45 	imes 2^9 + 10 	imes 2^9 = 55 	imes 2^9$.Comparing with the statements:Statement $-1$ is $55 	imes 2^9$,which is true.Statement $-2$ claims $s_2 = 10 	imes 2^8$,but we found $s_2 = 10 	imes 2^9$,so Statement $-2$ is false.Thus,Statement $-1$ is true and Statement $-2$ is false.

Let $s_1 = \sum_{j=1}^{10} j(j-1) \binom{10}{j}$,$s_2 = \sum_{j=1}^{10} j \binom{10}{j}$,and $s_3 = \sum_{j=1}^{10} j^2 \binom{10}{j}$.
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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Let $s_1 = \sum_{j=1}^{10} j(j-1) \binom{10}{j}$,$s_2 = \sum_{j=1}^{10} j \binom{10}{j}$,and $s_3 = \sum_{j=1}^{10} j^2 \binom{10}{j}$.Statement $-1$: $s_3 = 55 \times 2^9$Statement $-2$: $s_1 = 90 \times 2^8$ and $s_2 = 10 \times 2^8$