Question

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

• A. 9
• B. 8
• C. 5
• D. 7

Answer 1

The correct option is B 8

$\frac{(n+1)n(n-1)}{6}-\frac{(n+1)n}{2}\le 100$
$(n+1)n(n-4)\le 600$
Also $n-2\ge 0=n\ge 2$
and $n-1\ge 0=n\ge 1$
$\Rightarrow n\ge 2$
$\therefore$ Possible values are $n=2,3,4,5,6,7,8,9$