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10 Aug 2011, 00:39
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Q: Count number of ways to arrange 4 people A, B, C, D in a row so that C, D not sit next to each other
[*] Manhattan solution: (Total number of arrangement - number C, D sit next)
Manhattan approach: pretend that C, D stuck together : then Count the number of ways 2 people not sitting next to each other,
Total number of arrangements: 4! = 24
then number of ways of arrangement so that C, D next to each other is: 3! = 6. Since C, D are distinct, so all number of ways C, D next to each other is: 6x2 = 12.
--> Number of arrangements C, D not sit next: 24 -12 = 12
[*]PFD: (Total number of arrangement - number C, D sit next)
PFD approach: Force C sit on D's right by creating 3 slots for D to sit in: seat 1,2,3. After D choose his seat, S automatically sit on his right.
So, total number of arrangement where C, D next to each other: 3x1 = 3 ( since D can sit on C's right --> then total arrangement is: 3x2 = 6
---> Number of arrangement C, D not sit next: 24 - 6 = 18 (page 83 PFD)
Please help to find the error here? why two answer different
[*] Manhattan solution: (Total number of arrangement - number C, D sit next)
Manhattan approach: pretend that C, D stuck together : then Count the number of ways 2 people not sitting next to each other,
Total number of arrangements: 4! = 24
then number of ways of arrangement so that C, D next to each other is: 3! = 6. Since C, D are distinct, so all number of ways C, D next to each other is: 6x2 = 12.
--> Number of arrangements C, D not sit next: 24 -12 = 12
[*]PFD: (Total number of arrangement - number C, D sit next)
PFD approach: Force C sit on D's right by creating 3 slots for D to sit in: seat 1,2,3. After D choose his seat, S automatically sit on his right.
So, total number of arrangement where C, D next to each other: 3x1 = 3 ( since D can sit on C's right --> then total arrangement is: 3x2 = 6
---> Number of arrangement C, D not sit next: 24 - 6 = 18 (page 83 PFD)
Please help to find the error here? why two answer different
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10 Aug 2011, 13:04
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tranglenh
The manhattan way would be the way I do it, and the answer would be 12. I'd imagine either the second way is wrong or it's another problem?
but let's just humor their approach
D C _ _
_ D C _
_ _ D C
then
C D _ _
_ C D _
_ _ C D
looks like 6 ways C D are adjacent, but it doesn't account for A B swapping spots in each of them.
D C _ _
D C A B
D C B A
_ D C _
A D C B
B D C A
etc. so you;'re missing half the possibilities already. Double the 6 they originally claim you get 12. 24 total possibilities (4!) minus 12 = 12.
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Quantitative Questions Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
