Question

The number of positive integer satisfying the inequality ${}^{n+1}{C}_{n-2}{-}^{n+1}{C}_{n-1}\le 100$ is

• A. 9
• B. 8
• C. 5
• D. None of these

Answer 1

Answer: (B)

${}^{n+1}{C}_{n-2}-{}^{n+1}{C}_{n-1}\le 100$
$\Rightarrow {}^{n+1}{C}_3-{}^{n+1}{C}_2\le 100$
$\Rightarrow \frac{(n+1)n(n-1)}{6}-\frac{(n+1)n}{2}\le 100$
$\Rightarrow (n+1)n(n-1)-3n(n+1)\le 600$
$\Rightarrow (n+1)n(n-4)\le 600$
The values of $n$ satisfying this inequality are
$2,3,4,5,6,7,8,9$