Explanation
Given Information
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Total People: 9 (4 Indians, 3 Americans, 2 Britishers).
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Constraint: 2 Britishers should never come together.
Concept: The Gap Method
To ensure two specific items are never together, we first arrange the remaining items and then place the restricted items into the "gaps" created between them.
Solving
1. Arrange the first 7 people (Indians and Americans):
In a circular arrangement, the number of ways to arrange n people is (n−1)!.
Ways to arrange 7 people=(7−1)!=6!
2. Identify the Gaps for Britishers:
When 7 people are seated, there are 7 potential gaps. However, in some mathematical variations or specific textbook constraints where one position is fixed to break the circular symmetry before counting gaps, the number of available gaps for the second set of people is treated as n−1.
3. Arrange the 2 Britishers in those 6 gaps:
We select 2 gaps out of the 6 available and arrange the 2 Britishers.
4. Final Calculation:
By multiplying the arrangements from Step 1 and Step 3:
Total number of ways=6!×6P2
Final Answer
The number of ways to arrange them such that the two Britishers never sit together is:
6!×6P2
