Question

Assertion :$\sum _{r=2}^{10}{{}^{r}C}_2.{{}^{10}C}_r=11520$ and Reason: ${{}^{n}C}_0+{{}^{n}C}_1+{{}^{n}C}_2+......+{{}^{n}C}_n=2^n$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is correct Reason is incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$\sum _{r=2}^{10}{{}^{r}C}_2.{{}^{10}C}_r=\sum _{r=2}^{10}\frac{r(r-1)}{2}.\frac{10!}{r!(10-r)!}=\frac{90}{2}\sum _{r=2}^{10}{{}^{8}C}_{r-2}$
$=\frac{90}{2}({{}^{8}C}_0+{{}^{8}C}_1+....{{}^{8}C}_8)$
$=\frac{90\times 2^8}{2}=11520$.

Hence, both assertion and reason are correct and reason is correct explanation for assertion.