Question
Let $${S_1} = \sum\limits_{j = 1}^{10} {j\left( {j - 1} \right)\,{\,^{10}}{C_j},\,\,{S_2} = \sum\limits_{j = 1}^{10} {j\,{\,^{10}}{C_j}\,{\text{and }}{S_3} = \sum\limits_{j = 1}^{10} {{j^2}\,{\,^{10}}{C_j}.} } } $$
Statement - 1 : $${S_3} = 55 \times {2^9}.$$
Statement - 2 : $${S_1} = 90 \times {2^8}\,{\text{and }}{S_2} = 10 \times {2^8}.$$
A.
Statement - 1 is true, Statement - 2 is true ; Statement - 2 is not a correct explanation for Statement - 1.
B.
Statement - 1 is true, Statement - 2 is false.
C.
Statement - 1 is false, Statement - 2 is true.
D.
Statement - 1 is true, Statement - 2 is true ; Statement - 2 is a correct explanation for Statement - 1.
Answer :
Statement - 1 is true, Statement - 2 is false.
Solution :
$$\eqalign{
& {S_2} = \sum\limits_{j = 1}^{10} {j\,{\,^{10}}{C_j} = \sum\limits_{j = 1}^{10} {10} \,{\,^9}{C_{j - 1}}} \cr
& = 10\left[ {^9{C_0} + {\,^9}{C_1} + {\,^9}{C_2} + ..... + {\,^9}{C_9}} \right] = 10.\,{2^9} \cr} $$