Question

Assertion :The expression ${}^{40}{C_r}^{60}{C}_0{+}^{40}{C_{r-1}}^{60}{C}_1+...$ attains maximum value when $r=50.$ Reason: ${}^{2n}{C}_r$ is maximum when $r=n$

• 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 incorrect and Reason is incorrect

Answer 1

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

We have

$\frac{\left({}^{2n}Cr\right)}{\left({}^{2n}{C}_{r+1}\right)}=\frac{\left(2n\right)!}{r!(2n-r)!}\frac{(r+1)!(2n-r-1)!}{\left(2n\right)!}=\frac{r+1}{2n-r}$

Since for $0\le r\le n-1,\frac{r+1}{2n-r}<1$, we get

$\left({}^{2n}{C}_0\right)<\left({}^{2n}{C}_1\right)<...<\left({}^{2n}{C}_{n-r}\right)<\left({}^{2n}{C}_n\right)$

Also, as $\left({}^{2n}{C}_r\right)=\left({}^{2n}{C}_{2n-r}\right)$

$\left({}^{2n}{C}_{2n}\right)<\left({}^{2n}{C}_{2n-1}\right)<...<\left({}^{2n}{C}_{n+1}\right)<\left({}^{2n}{C}_n\right)$

Thus, $\left({}^{2n}{C}_r\right)$ is maximum when $r=n$

Next, $\left({}^{40}{C}_r\right)\left({}^{60}{C}_0\right)+\left({}^{40}{C}_{r-1}\right)\left({}^{60}{C}_1\right)+...=$

The number ways of selecting $r$ persons out of $40$ men and $60$ women $=\left({}^{100}{C}_r\right)$

Which is maximum when $r=50$.