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Question Number 64414 by Yitagesuweldesenbet@gmail. Com last updated on 17/Jul/19

The number of positive integers satisfying the inequality ^(n+1) C_(n−2) −^(n+1) C_(n−1) ≤ 100 is ____.

Thenumberofpositiveintegerssatisfyingtheinequalityn+1Cn−2−n+1Cn−1⩽100is____.

Answered by mr W last updated on 17/Jul/19

^(n+1) C_(n−2) =^(n+1) C_3 =(((n+1)n(n−1))/(3!)) ^(n+1) C_(n−1) =^(n+1) C_2 =(((n+1)n)/(2!)) (((n+1)n(n−1))/(3!))−(((n+1)n)/(2!))≤100 (n+1)n(n−4)≤600 with n=9: LHD=10×9×5=450<600 with n=10: LHD=11×10×6=660>600 ⇒2≤n≤9

n+1Cn−2=n+1C3=(n+1)n(n−1)3!n+1Cn−1=n+1C2=(n+1)n2!(n+1)n(n−1)3!−(n+1)n2!⩽100(n+1)n(n−4)⩽600withn=9:LHD=10×9×5=450<600withn=10:LHD=11×10×6=660>600⇒2⩽n⩽9

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