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Seven seats are arranged in a row. Three persons are to be seated such that the middle seat (seat 4) is always occupied and no two of the three persons sit in adjacent seats. The number of ways this can be done equals n!. Using the value of n so found, determine which of the following statements are true.
- The number of ways to choose n points from 12 distinct collinear points such that no two chosen points are consecutive is 9C5
- The number of (n+1)-digit numbers in which every digit is strictly greater than the digit immediately to its left is 9C5
- The number of odd proper divisors of 20412 equals 3n + 1
- The inequality 1/2 + log₆(n) + log₈(6) >= 3^(2/3) holds
Correct answer: The number of ways to choose n points from 12 distinct collinear points such that no two chosen points are consecutive is 9C5
Solution
With seat 4 fixed and seats 3,5 excluded, the 2 remaining persons choose from {1,2,6,7} with no two adjacent. Valid pairs: (1,6),(1,7),(2,6),(2,7) giving 4 choices. Total ways = 4 * 3! = 24 = 4!, so n=4. Statements A, B, and C are all true for n=4.
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