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- Determine the number of arrangements where 6 boys and 3 girls can be seated in a row, ensuring at least two girls sit together, while the boys can be in any order.<stron
Determine the number of arrangements where 6 boys and 3 girls can be seated in a row, ensuring at least two girls sit together, while the boys can be in any order.
9652
1000
9604
5208
Answers (1)
To determine the number of arrangements where 6 boys and 3 girls can be seated in a row, ensuring at least two girls sit together, we can consider the following cases:
Case 1: Two girls sit together:
In this case, we can treat the two girls who sit together as a single entity. This reduces the problem to arranging 7 entities: the combined entity of the two girls, the remaining girl, and the 6 boys. These 7 entities can be arranged in ways.
Case 2: All three girls sit together:
In this case, we treat the three girls who sit together as a single entity. This reduces the problem to arranging 4 entities: the combined entity of the three girls and the 6 boys. These 4 entities can be arranged in ways.
Case 3: Two pairs of girls sit together:
In this case, we can consider the two pairs of girls as single entities. This reduces the problem to arranging 5 entities: the combined entity of the two pairs of girls, the remaining girl, and the 6 boys. These 5 entities can be arranged in ways.
Case 4: All three girls sit together and one girl sits separately:
In this case, we treat the three girls who sit together as a single entity and the remaining girl as a separate entity. This reduces the problem to arranging 4 entities: the combined entity of the three girls, the remaining girl, and the 6 boys. These 4 entities can be arranged in ways.
Total number of arrangements = (Number of arrangements in Case 1) + (Number of arrangements in Case 2) + (Number of arrangements in Case 3) + (Number of arrangements in Case 4)
= 5,040 + 24 + 120 + 24
= 5,208
Therefore, there are 5,208 different arrangements where 6 boys and 3 girls can be seated in a row, ensuring at least two girls sit together, while the boys can be in any order.
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