Question

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 $S_2=10\times 2^8$.

• A. Statement-1 is true, Statement-2 is true; Statement-2 is not the 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 the correct explanation for Statement-1

Answer 1

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

$S_1={\sum }_{j=1}^{10}|(j-1)\frac{10!}{(j-1)(j-2)!(10-j)!}=90{\sum }_{j=2}^{10}\frac{8!}{(j-2)!(8-(j-2\left)\right)!}=90\cdot 2^8$
$S_2={\sum }_{j=1}^{10}|\frac{10!}{j(j-1)!(9-(j-1\left)\right)!}=10{\sum }_{j=1}^{10}\frac{9!}{(j-1)!(9-(j-1\left)\right)!}=10\cdot 2^9$
$S_3=\sum _{j=1}^{10}\left[j\right(j-1)+j]\frac{10!}{|!(10-j)!}=\sum _{j=1}^{10}|(j-1{)}^{10}{C}_j=\sum _{j=1}^{10}{j}^{10}{C}_j={90.2}^8+{10.2}^9$
$={90.2}^8+{20.2}^8={110.2}^8={55.2}^9$.