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

The correct option is B $8$

${{}^{n+1}C}_{n-2}-{{}^{n+1}C}_{n-1}\le 50$
$\Rightarrow \frac{(n-1)!}{3!(n-2)!}-\frac{(n+1)!}{2!(n-2)!}\le 50$
$\Rightarrow (n+1)!\left(\frac{n-1-3}{(n-1)!}\right)\le 300$
$\Rightarrow (n+1)n(n-4)\le 300$

By trail and error we get $n\le 8$ to satisfy the above inequality.