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 ]