Question

The sum of the products of the ten numbers $\pm 1,\pm 2,\pm 3,\pm 4,\pm 5$ taking two at a time is

• A. $165$
• B. $-55$
• C. $55$
• D. none of these

Answer 1

The correct option is B $-55$

Let $S$ is sum of the products of the ten numbers $\pm 1,\pm 2,\pm 3,\pm 4,\pm 5$ taking two at a time.
Therefore, $S=1(-1)+2(-2)+3(-3)+4(-4)+5(-5)$

$\Rightarrow S=-{\sum }_{r=1}^5{r}^2$

$=-\frac{5(5+1)(10+1)}{6}$ [ By using ${\sum }_{i=1}^n{k}^2=1^2+2^2+3^2+\dots +n^2=\frac{n(n+1)(2n+1)}{6}$]

$\Rightarrow S=-55$
Ans: B