Question

If ${}^7{C}_r\ge {3.}^7{C}_{r-1}$, then minimum value or $r$ is

• A. $2$
• B. $3$
• C. $4$
• D. $6$

Answer 1

The correct option is A $2$

${}^7{C}_r\ge 3{\Rightarrow }^7{C}_{r-1}\Rightarrow \frac{7!}{(7-r)!r!}\ge 3x\frac{7!}{(7-r+1)!(r-1)!}\Rightarrow \frac{7!}{(7-r)!(r-1)!}[\frac{1}{r}-\frac{3}{7-r+1}]\ge 0\Rightarrow \frac{7!}{(7-r)!(r-1)!}\left[\frac{7-r+1-3r}{\left(r\right)(7-r+1)}\right]\ge 0\Rightarrow \frac{7!(8-4r)}{(7-r+1)!\left(r\right)!}\ge 0\Rightarrow \frac{7!\left(4\right)(2-r)}{(8-r)!\left(r\right)!}\ge 0\Rightarrow 2-r\ge 0\Rightarrow r=2$

It is the least value

Hence the correct answer is $2$