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Seating in a Movie Theater How many different ways can 5 people- $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D},$ and $\mathrm{E}-$ sit in a row at a movie theater if $(a) \mathrm{A}$ and $\mathrm{B}$ must sit together; $(b) \mathrm{C}$ must sit to the right of, but not necessarily next to, $\mathrm{B} ;(c) \mathrm{D}$ and $\mathrm{E}$ will not sit next to each other?

a. 48 b. 60 c. 72

Intro Stats / AP Statistics

Chapter 4

Probability and Counting Rules

Section 4

Counting Rules

Sampling and Data

Probability Topics

Temple University

Piedmont College

Cairn University

University of St. Thomas

Lectures

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five people A, B, C, D and E want to sit in a row at the movie theater? In how many ways can this be done? If a A and B want to sit together, B. C must sit to the right to be, but not necessarily next to be see the any. Don't want to sit next to each other. Let's look at the first question so A and B want to sit together in order to our answer. This question. What we want to do is to think off A and B s one person a b right seated in that order, a then b So are four people would be a B see Dean. And so we have four people that we want to sit in in a role for possibilities for the first seat, three for the next two for the third and one for the last. We multiply those numbers because of the fundamental counting rule. This gives is for a factorial, which is equal to 24 but A and B can be seated in the order be then a So are four people could be B A C. Dean and and there would still be four factorial ways of seating them. So the total number of ways that we could sit the five people, if A and B want to sit together will be the 24 place 24 equal to 48 waste for the next question. Green. In how many ways can we see the five people IFC must sit to the right of B. In this case, we look at the difference in our years. In each scenario, we're going toe fix the position of being okay. So let's look at the case where B sits in the first seat. So there's one possibility for that seat. Just be is going to sit there for the second seat. Any of the other four can sit there, including C for the third seat. Three possibilities for two seats, too. Last seat one two. Under the scenario, there's four factorial ways to see them or 24. Let's put B in the second seat, so one possibility for that seat for the first seat thank be, won't sit there, and C cannot sit there because it is to the left to be so there will only be three parts abilities. A D or E No for the third seat. Two have already been seated. C can sit there so there will be three possibilities here. Fort seek to last seat one fundamental counting rule. We just multiply those numbers and we get 18. What if we put B in the third seat? And so one possibility for the third seat for that first seat B cannot sit there. See? Cannot sit there because it is to the left of B. All right, so there will only be three possibilities a d and E in the second seat. Only two possibilities see, cannot sit there because again, it's still to the left of B for the fourth seat. We know seeking can sit there and and three had already been seated. So only two possibilities for the last seat. One possibility by the fundamental counting principle or rule, we just multiply these numbers. That gives us 12 next weekend. Put B in the fort seat name. So one possibility for the fourth seat just being now to the very first seat there will be only three possibilities because be has been seated and see cannot sit there so three possibilities. Similarly, second seat, there will only be two possibilities. Third seat one. And on the fifth seat, there will only be one possibility. If we multiply these numbers by the fundamental counting rule, we get six. Note that be cannot sit on the fifth seat because then see won't be able to sit anywhere because there is no way see can be seated to the right to be. So These are all the scenarios. These are the only scenarios that that we need to look at in the total number of ways to see the five people would be 24 plus 18 plus 12 plus six. That gives you a total of 60 ways for questions. See, in how many ways can we sit five people in a role if d and E. I don't want to sit next to each other? Shall we know that if we want to sit five people with no conditions or restrictions, there would be five times four times, three times, two times one five factorial ways of doing this. That's 120 from our answer in a If two particular people want to sit next to each other, right, if they want to sit together, and there are 48 ways of doing that to this is if any two particular people, like D and E said next to each other. So the number of ways the five people can be seated where D and E are not seated next to each other. It's just going to be the difference between these two. So 1 20 minus 48. So you take away the number of ways they can be seated. If d and sit next to each other, they should give you the answer to this question, which is going to be 72. This is the number of ways to fight. People can be seated if D and E don't want to sit next to each other.

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