Question

Let $S_1={\sum }_{j=1}^{10}j(j-1){}^{10}{C}_j,S_2={\sum }_{j=1}^{10}j.{}^{10}{C}_j,S_3={\sum }_{j=1}^{10}{j}^2{.}^{10}{C}_j.$
Statement 1: $S_3=55\times 2^9.$
Statement 2: $S_1=90\times 2^{5}and{S}_2=10\times 2^8$

• A. Statement 1 is false, statement 2 is true.
  
• B. Statement 1 is true, statement 2 is true: statement 2 is a correct explanation for statement 1.
• C. Statement 1 is true, statement is true: statement 2 is not a correct explanation for statement 2.
• D. Statement 1 is true statement 2 is false.

Answer 1

The correct option is D Statement 1 is true statement 2 is false.

$S_1={\sum }_{j=1}^{10}j(j-1)\frac{10!}{j(j-1)(j-2)!(10-j)!}=90{\sum }_{j=2}^{10}\frac{8!}{(j-2)!(8-(j-2\left)\right)!}=90{\sum }_{j=2}^{10}{}^8{C}_{j-2}=90\times 2^8{S}_2={\sum }_{j=1}^{10}\frac{j.10!}{j(j-1)!(9-(j-1\left)\right)!}=10{\sum }_{j=1}^{10}\frac{9!}{(j-1)!(9-(j-1\left)\right)!}=10{\sum }_{j=1}^{10}{}^9{C}_{j-1}=10\times 2^9{S}_3=\sum \left[j\right(j-1)+j]{}^{10}{C}_J={\sum }_{j=1}^{10}j(j-1){}^{10}{C}_j+{\sum }_{j=1}^{10}j{.}^{10}{C}_j=90\times 2^8+10\times 2^9=55\times 2^9$