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  • A group of 12 friends, including 5 men and 7 women, is going to sit at a rectangular table. If the men and women must alternate seats, and the two men and two women must

A group of 12 friends, including 5 men and 7 women, is going to sit at a rectangular table. If the men and women must alternate seats, and the two men and two women must sit at the corners, in how many different ways can they be seated?

 

Option: 1

2560

 


Option: 2

6052

 


Option: 3

5682

 


Option: 4

5760


Answers (1)

best_answer

To calculate the number of different ways the group of 12 friends can be seated at a rectangular table, with the men and women alternating seats and the two men and two women sitting at the corners, we can consider the following:

Let's first arrange the two men and two women at the corners. There are 2 ! ways to arrange the men at the corners and 2 ! ways to arrange the women at the corners. Therefore, there are 2 ! \times 2 !=4 different arrangements for the corners.

Now, we have 8 friends remaining: 3 men and 5 women.

We need to arrange these 8 friends in the remaining 8 seats, with the constraint that men and women must alternate seats.

We can treat the men as a single entity and the women as a single entity. Therefore, we have 2 entities to arrange: the combined entity of the 3 men and the combined entity of the 5 women.

The number of ways to arrange these 2 entities is 2 !

Within the combined entity of the 3 men, the men themselves can be arranged among themselves, which gives us 3 ! possibilities.

Therefore, the total number of different ways to arrange the group of 12 friends at a rectangular table, with the men and women alternating seats and the two men and two women sitting at the corners, is:

4 \times 2 ! \times 3 ! \times 5 !=4 \times 2 \times 6 \times 120=5,760 .

Therefore, there are 5,760 different ways the group of 12 friends can be seated at a rectangular table, satisfying the given conditions.

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