Question

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

• A. 9
• B. 8
• C. 7
• D. 6

Answer 1

Answer: (B)

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