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# Combinatorics/Permutations Fundamental Question

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Intern
Joined: 06 Aug 2018
Posts: 8

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25 Aug 2018, 15:11
1
Hello All,

When dealing with permutation questions how do you know when to take the fundamental counting principle approach versus taking the total number of arrangements - unacceptable arrangements?

if we take the example:

There are 6 people, A, B, C, D, E and F. They have to sit around a circular table such that A can sit neither next to D nor next to F. How many such arrangements are possible?

The approach I would take is:

Total arrangements = (6-1)! = 5!

Unacceptable arrangements = AF,FA, DA and AD
= 2*4! + 2*4
= 4*4!

but we have accounted for FAD twice in counting arrangements for FA & AD and the same for DAF. So we need to remove this from our unacceptable arrangements.
= 4*4! - 2*3!

Acceptable arrangements = 5! - (4*4! - 2*3!)
= 120 - (24*4 - 12)
= 120 - (96 - 12)
= 36

But the much simpler approach would be to calculate the following below using the attached diagram

acceptable arrangements = 1*3*1*2*3*2*1
= 36
Attachments

Capture.PNG [ 7.81 KiB | Viewed 620 times ]

Senior Manager
Joined: 18 Jun 2018
Posts: 262

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27 Aug 2018, 04:49
1
Thanks for sharing your approach. Really helpful in solving such type of questions.
Re: Combinatorics/Permutations Fundamental Question   [#permalink] 27 Aug 2018, 04:49
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