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

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)

thanks..

_________________
Tim Sanders
Manhattan GMAT Instructor

Got it. Thanks!

:)

_________________
Tim Sanders
Manhattan GMAT Instructor