Question

Let $S_1={\sum }_{j=1}^{10}j(j-1{)}^{10}{C}_j,S_2={\sum }_{j=1}^{10}{j}^{10}{C}_j\text{and}{S}_3={\sum }_{j=1}^{10}{j}^2{}^{10}{C}_j.$
Statement -1 $S_3=55\times 2^9$
Statement-2 $S_1=90\times 2^8\text{and}{S}_2=10\times 2^8$

• A. Statement-1 is true, Statement-2 is true: Statemnt-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 false.
  
• D. Statement-1 is true, Statement 2 is true: Statement -2 is a correct explanation for Statement-1.

Answer 1

The correct option is B Statement -1 is true, Statement-2 is false .


$S_2={\sum }_{j=1}^{10}{j}^{10}{C}_j={\sum }_{j=1}^{10}10{{}^{9}C}_{j-1}=10[{}^9{C}_0{+}^9{C}_1{+}^9{C}_2+...{+}^9{C}_9]={10.2}^9$