Explanation

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