Required Mathematical Intelligence

Chapter 102 Husband and Wife

Someone invited three couples to lunch, and arranged for everyone (including the host himself and his wife) to sit around a round table, so that the men and women could be mixed without any husband sitting next to his wife.

Question: How many ways can you sit like this?If you only pay attention to the order of each seat, and do not count the number of ways to sit in the same order but in different places.

[Answer: The husbands are seated, and their wives are placed beside each of them, there are obviously 6 ways of sitting (not 24, since we are only considering the order of the positions).Now, let each husband stay in his place, move the first lady to the second seat, the second lady to the third seat, and so on until the fourth seat , and change the fourth lady to the first position.This way of sitting meets the requirements of the title, that is, the husband does not sit next to his wife.There are also 6 kinds of this sitting method, each of which can make the wife continue to move forward by one position, and this just gets 6 kinds of feasible schemes again.But it is impossible to get the wives to change seats, otherwise, the wives should sit with their husbands, just in a different direction.

Therefore, there are a total of 6 6 = 12 possible seating schemes.Below we use Roman numerals (from I to IV) to represent the husband, and Arabic numerals to represent the wife (also 1 to 4), so that everything is clear.The first 6 permutations are:

Ⅰ4 Ⅱ1 Ⅲ2 Ⅳ3
Ⅰ3 Ⅱ4 Ⅲ1 Ⅳ2
Ⅰ2 Ⅲ1 Ⅳ3 Ⅱ4
Ⅰ4 Ⅲ2 Ⅳ1 Ⅱ3
Ⅰ3 Ⅳ1 Ⅱ4 Ⅲ2
Ⅰ2 Ⅳ3 Ⅱ1 Ⅲ4
The other 6 arrangements are the same, except that the order of the positions of men and women is reversed. ]
(End of this chapter)