Problem

Seating in a Movie Theater How many different way…

08:40

Problem 68 Hard Difficulty

People Seated in a Circle In how many ways can 3 people be seated in a circle? 4? n? (Hint: Think of them standing in a line before they sit down and/or draw diagrams.)

Answer

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Intro Stats / AP Statistics

Elementary Statistics a Step by Step Approach

Chapter 4

Probability and Counting Rules

Section 4

Counting Rules

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Video Transcript

three people are to be seated in a circle In how many ways can this be done? What did there are? Four people. What if there are any people? Let's look at the first question. Let's say we have three people a B c Seated in a circle That's a sign names to do seats 12 and three. And suppose I look at on arrangement based on the first arrangement where I just moved everyone one seat in a clockwise direction so that they will be in Seat two, Basin three, season one. I know that these two arrangements are actually this same because each person has the same position with respect to the others. For example, if you look at game in the first arrangement to his right to see this left is being in the second arrangement. If you look at a please write this scene, this left this being and the same is true for the other people being seen. So we say this to arrangements are actually the same. Let's say, have another arrangement and what I'm gonna do, same gonna fix a in the first seat going to put him there and is going to be fixed in in seat number one and then we switch the positions of B and C, so that be seen thing. Season two. This arrangement is different from the first. If you look at a this right, this be this left, it's seen. So we look and look at the analysis in a linear fashion. This air seat number, swan two and three. What we do is we fixed the position off one of them, right? So in seat number one, we assign one person to sit there. Then we find the number off possibilities for seats two and three. There will be two possibilities for number two. One possibility for seat number three. By the fundamental counting room, there will be two factorial ways for two ways off. Arranging three people in a circular table. Let's look at the next question. Suppose there are four people. Hey, let's look at the analysis in this linear freshen. Is there seat numbers? 123 and four. We're going to fix the position off one of them. Put that person in seat number one in seat number two. There will be three possibilities. Any of the remaining three people seat number 32 Possibilities. Seat number four one possibility by the fundamental counting rule. We just want to play those numbers and get three factorial or six different ways of seating for people in a circle. Suppose there are in people. We used the same analyses. I mean, we have in seat in all for the first seat, we fixed the position off one of the end people for the second seat. You need the remaining in minus one can sit there and so on For the end seat, there would be one possibility by the fundamental counting rule. We just want to apply those numbers and the answer would be in minus one factorial ways of seating in people in a circle.

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