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Difficulty: HardPermutations, Combinations, and Fundamental Counting Principle

A project manager is scheduling 77 distinct project milestones: 44 technical milestones and 33 managerial milestones. The milestones must be scheduled sequentially across 77 consecutive weeks. To avoid scheduling burnout, no two managerial milestones can be scheduled in consecutive weeks. In how many different valid sequences can all 77 milestones be scheduled?

  1. A
    240240
  2. B
    720720
  3. 1,4401,440Answer
  4. D
    2,1602,160
  5. E
    5,0405,040

Answer

1,4401,440
To ensure no two managerial milestones are adjacent, first order the 44 distinct technical milestones, which can be done in 4!=244! = 24 ways. These 44 milestones create 55 available slots (gaps before, between, and after them). To place the 33 distinct managerial milestones into these 55 slots such that no slot contains more than one managerial milestone, we calculate the permutations P(5,3)=5×4×3=60P(5, 3) = 5 \times 4 \times 3 = 60. Multiplying the independent choices yields 24×60=1,44024 \times 60 = 1,440 valid sequences.

Step-by-Step Solution

1
Arrange the non-restricted items (the 4 distinct technical milestones)
Number of ways =4!=24= 4! = 24
Since all 4 technical milestones are distinct, they can be ordered in 4!4! ways.
2
Determine the number of available slots (gaps) created for the restricted items
Number of slots =4+1=5= 4 + 1 = 5
Placing 4 technical milestones in a line creates 5 potential slots (before the first, between adjacent ones, and after the last) where managerial milestones can be placed without being adjacent to each other: _ T1 _ T2 _ T3 _ T4 _
3
Place and order the 3 distinct managerial milestones into the 5 available slots
Number of ways =P(5,3)=5×4×3=60= P(5, 3) = 5 \times 4 \times 3 = 60
Because the managerial milestones are distinct and order matters, we choose 3 slots out of 5 and arrange them.
4
Apply the Fundamental Counting Principle to find the total valid arrangements
Total valid arrangements =24×60=1,440= 24 \times 60 = 1,440
The placement of technical milestones and managerial milestones are independent choices in sequence.

Key Concept

Permutations with Non-Adjacency Restrictions (Gap Insertion Method)
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Topics

Permutations, Combinations, and Fundamental Counting Principle

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