Question

Number of elements in set of values of $r$ for which ${}^{18}{C}_{r-2}+{2.}^{18}{C}_{r-1}+{}^{18}{C}_r\ge {}^{20}{C}_{13}$ is satisfied is:

• A. $4$
• B. $5$
• C. $7$
• D. $10$

Answer 1

Answer: (C)

$7$

${{}^{18}C}_{r-2}+2.{{}^{18}C}_{r-1}+{{}^{18}C}_r\ge {{}^{20}C}_{13}$

$\Rightarrow [{{}^{18}C}_{r-2}+{{}^{18}C}_{r-1}]+[{{}^{18}C}_{r-1}+{{}^{18}C}_6]\ge {{}^{20}C}_{13}$

${{}^{19}C}_{r-1}+{{}^{19}C}_r\ge {{}^{20}C}_{13}$

${{}^{20}C}_r\ge {{}^{20}C}_{13}$

So, $r\ge 13$

So, $r=13,14,15,16,17,18,19,20$

So, $7$ elements.

Hence, the answer is $7.$