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Difficulty: HardPermutations, Combinations, and Fundamental Counting Principle

A research committee must select a delegation of 66 members from a pool of 44 senior fellows and 44 junior analysts to sit around a circular conference table with 66 evenly spaced seats. The delegation must consist of exactly 33 senior fellows and 33 junior analysts. If no two senior fellows may sit in adjacent seats around the table, how many distinct delegation seating arrangements are possible? (Two seating arrangements are considered identical if one can be rotated to match the other.)

Answer: 192

Answer

The total number of distinct delegation seating arrangements is 192.
To find the total number of distinct delegation seating arrangements, we first determine how many ways the 6 delegates can be selected, and then multiply by the number of valid circular seating arrangements for those delegates.

1. Selection of Delegates:
- Ways to select 3 senior fellows from 4: (43)=4\binom{4}{3} = 4
- Ways to select 3 junior analysts from 4: (43)=4\binom{4}{3} = 4
- Total delegate combinations: 4×4=164 \times 4 = 16

2. Circular Seating Arrangements:
- With 3 seniors and 3 juniors at a 6-seat table, no two seniors can sit adjacent if and only if seniors and juniors alternate seats.
- Fix one senior fellow to eliminate rotational symmetry.
- The remaining 2 seniors can be arranged in 2!=22! = 2 ways.
- The 3 junior analysts can be arranged in the 3 intermediate seats in 3!=63! = 6 ways.
- Seating arrangements per delegation = 2×6=122 \times 6 = 12

3. Total Arrangements:
- Total = 16×12=19216 \times 12 = 192.

Step-by-Step Solution

1
Calculate combinations of senior fellows and junior analysts to form the 6-person delegation.
Number of ways to choose 3 seniors out of 4 is (43)=4\binom{4}{3} = 4. Number of ways to choose 3 juniors out of 4 is (43)=4\binom{4}{3} = 4. Total selection combinations = 4×4=164 \times 4 = 16.
Choosing members from distinct pools uses combinations because member order within the selection does not matter.
2
Analyze the seating constraint for 3 seniors and 3 juniors around a 6-seat circular table.
The senior fellows must occupy alternating seats around the table (e.g., seats 1, 3, 5), leaving seats 2, 4, 6 for the junior analysts.
Placing 3 seniors among 6 circular seats with no two adjacent forces seniors to occupy every second seat.
3
Calculate the number of distinct circular seating arrangements for any specific set of 6 selected people.
Arrangements = (3−1)!×3!=2×6=12(3 - 1)! \times 3! = 2 \times 6 = 12.
To account for rotational symmetry at a circular table, fix one senior fellow's seat. The remaining 2 senior fellows can be seated in 2!=22! = 2 ways, and the 3 junior analysts can be seated in 3!=63! = 6 ways in the remaining open seats.
4
Apply the Fundamental Counting Principle to combine selection and seating steps.
Total arrangements = 16 (selections)×12 (seating arrangements)=19216 \text{ (selections)} \times 12 \text{ (seating arrangements)} = 192.
Each of the 16 unique delegations can be seated around the circular table in 12 distinct relative orders.

Key Concept

Combinations and Circular Permutations with Adjacency Restrictions
Estimated Time:2m 0s
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Topics

Permutations, Combinations, and Fundamental Counting Principle

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