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Mia and Daniel are planning their wedding.
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They’re working on the seating plan for the top table at the reception.
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Their top table is a straight line with eight seats down one side.
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It needs to sit the bride and groom, the bride’s parents, the groom’s parents, the best man, and the maid of honor.
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Given that all couples need to sit next to each other and that the best man and maid of honor are not a couple, how many different ways are there for seating everyone on the top table?
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We have a few restrictions on how we seat each couple and the maid of honor and the best man.
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Let’s begin by considering the couples who are the bride and groom, the bride’s parents, and the groom’s parents as three units essentially.
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And we’re going to begin by working out the total number of ways of just seating these three couples.
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There are three ways of choosing the first couple to seat.
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There are two ways of choosing the second couple to seat and one way of choosing the third.
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And, of course, the product rule for counting or the counting principle says the total number of options is the product of these.
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It’s three times two times one, which is six.
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So we have six ways of seating the couples.
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But of course, each couple could sit in a different order.
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We could have the bride and groom or the groom then the bride.
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And if we think about it, there are two ways to seat the bride and groom, two ways to seat the bride’s parents, and two ways to seat the groom’s parents.
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Two times two times two is equal to eight, meaning that there are eight ways that each couple could sit next to each other.
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Bear in mind that this is for each of the six original ways of seating the couples.
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This means that the total number of possibilities when it comes to seating these is the product of these two sets of outcomes.
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It’s six times eight, which is 48.
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So we have 48 ways in total of seating those couples.
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And so now we move on to seating the best man and the maid of honor.
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We consider these individually because we’re told they’re not a couple, and therefore they don’t necessarily need to sit next to one another.
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And so if we think about the top table with our three couples already seated, he could sit at either end.
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But also he could sit at any point between the couples.
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And so there must be four options of chairs for him.
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Then, once the best man is seated, the maid of honor could sit at either end.
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But she could also sit between any of the couples and/or the best man, depending on where he’s located.
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And this must mean that there are five different ways of seating the maid of honor.
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Now that we’ve considered all the possible events, that is, seating the couples, seating the best man, and seating the maid of honor, we know that the fundamental counting principle tells us to find the product of these.
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That’s 48 times four times five, which is equal to 960.
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There are a total number of 960 different ways for seating everyone on the top table.
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Now, in fact, this isn’t the only method of answering this problem.
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We can alternatively just consider that there are five different groups; there are three couples and two individuals.
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And so we would say that there are five ways of choosing the first group to seat, four ways of choosing the second group, three ways of choosing the third, and so on, giving us a total of 120 different ways to arrange these five places.
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Then we go back to considering how the couples are arranged.
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We know that each couple could sit in a slightly different order.
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And so there are two times two times two, which is eight arrangements for our couples.
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Once again, the fundamental counting principle tells us that the total number of different ways for seating everyone is the product of these.
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It’s 120 times eight, which is once again 960.