Probability in WT

[trangle.nh]

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 differen

[mithunsam]

Manhattan is correct.

Infact, both are same approach. The calculation is wrong in the second solution. It should have been 3*2 = 6 instead of 3*1 =3. This is because, after D taking its slot, there are 2 empty seats, which could be occupied in 2 different ways.

The rest of the calculation should follow as below
6*2 = 12 (instead of 3*2=6)
24-12 =12 (instead of 24-6=18)

[tim]

thanks..

[trangle.nh]

Got it. Thanks!

[tim]

:)