For any permutation with multiple restrictions, it is often easier to consider how each position can be picked (the slotting method) rather than trying to figure out how to manipulate the permutation equation to match the situation given. This is an example of one of these situations – it is easy to imagine different acceptable permutations but difficult to see what that might mean to a generalized permutation formula.
Think of this instead as six “slots” that must be filled and use basic counting principles. Consider how many ways each can be filled, then multiply the different possibilities together.
The first seat can be filled by any of the six people, since there are not yet any restrictions.
The second seat can be filled by anyone except the other half of the first couple. This means it can be filled 4 ways.
The third seat can then be filled by either of the two members of the last couple.
To find the total number of ways the first row can be arranged, just multiply these three numbers together to get:
6 x 4 x 2 = 48
As you start the second row, notice that there are only three people left. The first “slot” of the second row can be filled by anyone except the other half of the couple in the first seat in the first row. This means that there are two possibilities as to how to fill this chair.
There is only one way to fill each of the remaining chairs. To see why, consider an example. If the three couples are just represented as Aa, Bb, and Cc, look at what happens if the first row is in the order ABC and the second row starts with b:
A B C
b _ _
The only possible place to put ‘c’ is in that second position if you want to avoid a column from having two members of the same couple. There is only 1 way to choose the second two seats and 2 total ways to pick the second row.
The total number of arrangements can then be found by multiplying the value found for the first row by the value found for the second row to get 48 x 2 = 96.
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