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Question

Difficulty: MediumPermutations

In how many different ways can 55 boys and 33 girls be seated in a straight row such that all 33 girls must sit together?

Answer: 4320 ways

Answer

The total number of ways to seat 55 boys and 33 girls in a row such that all 33 girls sit together is 43204320.
To arrange 55 boys and 33 girls so that the girls are always together, treat the 33 girls as 11 single unit. Combined with the 55 boys, there are 66 units to arrange in a straight line, which can be done in 6!=7206! = 720 ways. Within their group, the 33 girls can be arranged in 3!=63! = 6 ways. By the multiplication principle, the total number of seating arrangements is 720×6=4320720 \times 6 = 4320.

Step-by-Step Solution

1
Group the restricted items into a single block
The 33 girls form 11 unit. Combined with the 55 boys, there are 5+1=65 + 1 = 6 units to arrange.
Since all 33 girls must sit together, treating them as a single block ensures they are not separated.
2
Calculate the linear arrangements of the combined units
The 66 units can be arranged in 6!=7206! = 720 ways.
The number of distinct ways to arrange nn items in a line is n!n!.
3
Calculate internal arrangements of the girls' block
The 33 girls can be arranged among themselves in 3!=63! = 6 ways.
The 33 girls inside the block are distinct individuals and can swap positions.
4
Apply the fundamental counting principle
Total arrangements = 6!×3!=720×6=43206! \times 3! = 720 \times 6 = 4320.
The total number of arrangements is the product of external block arrangements and internal block arrangements.

Key Concept

Permutations with grouping constraints (string method)
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Topics

Permutations

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