The number of positive integers satisfying the In-equality ${^{n+1}C}_{n-2}-{^{n+1}C}_{n-1} ≤100$, is _____.

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We have, ${^{n+1}C}_{n-2}-{^{n+1}C}_{n-1} ≤100$ $⇒{^{n+1}C}_3-{^{n+1}C}_2≤100$   $[∵{^nC}_r={^nC}{n_r}]$ $⇒\frac{(n+1)n(n-1)}{6}-\frac{(n+1)n}{2}≤100$ $⇒(n+1)n(n-1)-3n(n+1)≤600$ $⇒(n+1)(n)(n-4)≤600$ The values of n satisfying this inequality are 2,3,4,5,6,7,8,9.