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Difficulty: Very hardPermutations and Linear Arrangements

Eight executive team members—3 vice presidents, 3 directors, and 2 managers—are to stand in a single line for a company photograph. If no two vice presidents can stand next to each other, and the 2 managers must stand next to each other, how many different linear arrangements of the eight team members are possible?

Answer: 2880

Answer

2,880
To satisfy both constraints simultaneously, first treat the 2 managers as a single block, which has 2!=22! = 2 internal arrangements. Next, arrange the non-vice-president units—consisting of the 3 distinct directors and the 1 manager block—giving 4 units total, which can be ordered in 4!=244! = 24 ways. These 4 units create 5 distinct slot locations (at the ends and between adjacent units). To ensure no two vice presidents are adjacent, place the 3 distinct vice presidents into 3 of these 5 slots in P(5,3)=60P(5, 3) = 60 ways. Multiplying these independent decisions yields 2×24×60=2,8802 \times 24 \times 60 = 2,880.

Step-by-Step Solution

1
Group the elements that must remain adjacent and calculate internal permutations
The 2 managers form 1 block with 2!=22! = 2 internal arrangements.
Since the 2 managers must stand next to each other, treating them as a single block ensures they are never separated.
2
Arrange all non-restricted base units in a line
The 3 directors and 1 manager block yield 4!=244! = 24 linear arrangements.
Establishing the sequence of non-vice-president units creates the fixed slots into which the vice presidents will later be inserted.
3
Calculate the available slot arrangements for the separated elements
4 base units create 5 available slots. Permuting 3 vice presidents into 5 slots yields P(5,3)=60P(5, 3) = 60 ways.
Placing at most one vice president per slot guarantees that no two vice presidents are placed adjacently.
4
Apply the Fundamental Counting Principle to determine total arrangements
Total arrangements = 2×24×60=2,8802 \times 24 \times 60 = 2,880.
The choices for internal block arrangement, base unit ordering, and slot placement are independent sequential events.

Key Concept

Linear Permutations with Simultaneous Grouping and Non-Adjacency Constraints
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Topics

Permutations and Linear Arrangements

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