Question

The number of positive integers satisfying the inequality ${ }^{n+1} C_{n-2}-{ }^{n+1} C_{n-1} \leq 50$ is

• A. 9
• B. 8
• C. 7
• D. 6

Answer 1

Answer: (B)

Given, ${ }^{n+1} C_{n-2}-{ }^{n+1} C_{n-1} \leq 50$
$\Rightarrow \frac{(n-1) !}{3 !(n-2) !}-\frac{(n+1) !}{2 !(n-2) !} \leq 50 $
$\Rightarrow \frac{(n+1) !}{3 !}\left[\frac{1}{(n-2) !}-\frac{3}{(n-1) !}\right] \leq 50 $
$\Rightarrow(n+1) !\left(\frac{n-1-3}{(n-1) !}\right) \leq 300$
$\Rightarrow(n+1) n(n-4) \leq 300$
For $n=8$, it satisfy to the above inequality.
But $n=1$ it does not satisfy the above inequality.