Q117 / 170 · Manipal (MET) 2016 · Mathematics
The number of positive integers satisfying the inequality ${ }^{n+1} C_{n-2}-{ }^{n+1} C_{n-1} \leq 50$ is
- A$9$
- Correct answerB$8$
- C$7$
- D$6$
Solution
$\begin{aligned} & \text { Given, }{ }^{n+1} C_{n-2}-{ }^{n+1} C_{n-1} \leq 50 \\ & \Rightarrow \quad \frac{(n-1)!}{3!(n-2)!}-\frac{(n+1)!}{2!(n-2)!} \leq 50\end{aligned}$
$\Rightarrow \frac{(n+1)!}{3!}\left[\frac{1}{(n-2)!}-\frac{3}{(n-1)!}\right] \leq 50$
$\begin{array}{lr}\Rightarrow & (n+1)!\left(\frac{n-1-3}{(n-1)!}\right) \leq 300 \\ \Rightarrow & (n+1) n(n-4) \leq 300\end{array}$
For $n=8$, it satisfy to the above inequality.
But $n=1$ it does not satisfy the above inequality.
Hence, the correct answer is option: 'B'.