### Instruction: In how many ways can 7 people be seated in a row of chairs if two of the people, Wilma and Paul, refuse to sit next to each other?### Response: To count the number of ways to seat the 7 people, I will use the multiplication principle: the number of ways to do a series of tasks is the product of the number of ways to do each task.
First, I will assume that Wilma and Paul are willing to sit next to each other, and then I will subtract the cases where they do.
If Wilma and Paul can sit anywhere, then there are 7! ways to arrange the 7 people, because there are 7 choices for the first chair, 6 for the second, and so on.
Now, how many of these arrangements have Wilma and Paul sitting next to each other?
Well, I can think of them as a single unit, and then there are 6 units to arrange, so there are 6! ways to do that.
But within each unit, Wilma and Paul can switch places, so I have to multiply by 2 to account for that.
So there are 2 * 6! ways to have Wilma and Paul sitting next to each other.
Therefore, the number of ways to seat the 7 people without Wilma and Paul sitting next to each other is 7! - 2 * 6!.