#circular permutations question

86 messages · Page 1 of 1 (latest)

regal cloud
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There are five people sitting around a table—John, Jacob, Patrick, David, and Noah. In how many different ways can they sit around the table if Jacob and Patrick must be sitting side by side and David and Noah cannot be sitting side by side?

wispy birchBOT
ancient cypress
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Well, without restrictions there's 4! ways

eager barn
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yo broozeb please can u help me when u get a moment

silk citrus
eager barn
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oh ok

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i didnt know. im new bro

regal cloud
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I know that if one couple are sitting beside each other it’s 4!- 2!3!

static sparrow
ancient cypress
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No sorry, it's 4! ways on a circle

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And then if you glue say AB together then there's 2x(3!) ways I think

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Permutations on a circle of n people is (n-1)!

regal cloud
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Yeah when 2 people are sitting together it’s 12 ways

ripe scroll
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I would advise a subtractive method to handle the last restriction

regal cloud
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Yeah usually I use that method against the universal

ripe scroll
ancient cypress
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It feels like inclusion-exclusion except you're only excluding

ripe scroll
regal cloud
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For the first restriction I got 4! - 2!3! = 12

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and then two other people can’t sit beside each other

ancient cypress
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But does it make sense to subtract cases where two people are together? That's what you want

ripe scroll
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I don't follow

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Let's assume the question has no last restriction

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You think it's 12?

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yes i agree

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3!2!

regal cloud
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If there’s 5 people around a table and 2 must sit beside each other there are 4! - 3!2! Ways to arrange

ripe scroll
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Ok good. Now for the last restriction

ripe scroll
ancient cypress
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But that gives you the number of ways 5 people can sit around a table where the 2 arent sitting beside each other

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Because you're subtracting the cases where they are together

regal cloud
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I just take that there are now 4 people around the table and then u must permute the couple when they are sitting together so it’s 3!2!

ripe scroll
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what is 3!2!

regal cloud
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Hold on

ripe scroll
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Are you answering the question with no last restriction

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in which case, yes i agree

ancient cypress
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I think he means 2x(3!) , if you glue AB together then theres 4 objects on the circle, 3! ways to permute them, and also 3! ways to permute BA

regal cloud
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Oh ok ya

ripe scroll
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There are five people sitting around a table—John, Jacob, Patrick, David, and Noah. In how many different ways can they sit around the table if Jacob and Patrick must be sitting side by side~~ and David and Noah cannot be sitting side by side~~?

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So so far, the answer to this is 12, you agree right

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Then to finish the question, you think subtractively

regal cloud
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Yes

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That’s where I got stumped

ripe scroll
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Well what are you subtracting

regal cloud
ripe scroll
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yes

regal cloud
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So then I got like 3!2! And that just gives 0

ripe scroll
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There are five people sitting around a table—John, Jacob, Patrick, David, and Noah. In how many different ways can they sit around the table if Jacob and Patrick must be sitting side by sideand David and Noah cannot be sitting side by side and David and Noah ARE sitting side by side?

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no, how do u get 3!2!

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you need 2 pairs sitting next to each other

regal cloud
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2!2!

ripe scroll
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Its not that either

regal cloud
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2!2!2!

ripe scroll
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Let's rewind just to make sure

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How did you get 12 for the first part

regal cloud
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3!2!

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4 people around cuz the couple is just one

ripe scroll
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yes

regal cloud
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And we permute them

ripe scroll
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(4 - 1)!2!

regal cloud
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Ya

ripe scroll
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and hence the final answer

regal cloud
ripe scroll
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no no

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this is the subtractive method

regal cloud
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12 - 8 your saying is the final answer

ripe scroll
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Cases where 1 pair are together - Cases where 2 pairs are together = Cases where 1 pair is together, other pair is not

regal cloud
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Ok

ripe scroll
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I think we've given enough help - just have a think about it carefully

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to be sure

regal cloud
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Yeah this was a MC question from the exam I just took the reason why im confused was cuz the lowest answer was 8

ripe scroll
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its possible we've made an error

regal cloud
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Other options were 12 24 6

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I’ll try to figure it out though, thanks for the help sir

ripe scroll
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well one way to be sure is trying it manually...

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not many cases

ancient cypress
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That's probably what I'll end up doing

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It's kind of annoying to draw though

ripe scroll
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pong me if u get a non-4 answer

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hmm yh im convinced it is

ancient cypress
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I don't even know how to draw this actually

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Yeah, it’s 4

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