Six mobsters have arrived at the theater for the premiere of the film "Goodbuddies."
One of the mobsters, Frankie, is an informer, and he's afraid that another member
of his crew, Joey, is on to him. Frankie, wanting to keep Joey in his sights, insists
upon standing behind Joey in line at the concession stand, though not necessarily right
behind him. How many ways can the six arrange themselves in line such that Frankie’s
requirement is satisfied?
A. 6
B. 24
C. 120
D. 360
E. 720


I am little confused with the explanation given by MGMAT.
Suppose, I take the scenario like this -
1 2 3 4 5 6 -> 6 positions and A B C D J F -> 6 mobsters.
Now
If J is at position - 1, F can be in other 5 positions.
If J is at postion - 2, F can be in other 4 positions. [After J]
If J is at postion - 3, F can be in other 3 positions. [After J]
If J is at postion - 4, F can be in other 2 positions. [After J]
If J is at postion - 5, F can be in other 1 positions. [After J]
If J is at position - 6, F can not stand after that -> So we can consider the case that J can not stand at 6 position.
So the different combination is = 5 * 4 * 3 * 2 * 1 = 120.
So go for C. But why the OA is 360? Am I missing something ...

Solution-

if J takes Ist position, total no. of ways = 5 (available positions for F) X 4X3X2X1 (possible arrangements of the remaining 4)
if J takes IInd position, total no. of ways = 4 (available positions for F) X 4X3X2X1 (possible arrangements of the remaining 4)
similarly 3 X 4X3X2X1 or 2 X 4X3X2X1 or 1 X 4X3X2X1 possible ways for J's 3ed, 4th & 5th posotionings.

So total no. of ways = sum of all the above = (5+4+3+2+1) X (4X3X2X1) = 360

Ohhh! The Qs is asking for -


I missed that ... I thought in how many ways only J and F can arrange themselves and that's why I missed to arrange the rest 4 people.

Thanks idiot! Thanks a lot for guiding me the right path ...

Yes, nice explanation, "idiot"! Key words: possible arrangements of the remaining 4.

_________________
Emily Sledge
Instructor
ManhattanGMAT

This question can be done by the following method too.

Total number of ways 6 people can be arranged on a straight line = 6! = 6x5x4x3x2x1=720

In exactly half arrangements Joey will stand behind Frankie and in the other half, Frankie will stand behind Joey.

So the answer will be 720/2=360

I did it by this method. Hope you find it worthwhile.

Nice Pooja! That method exploits the symmetry of the arrangements. That shortcut is not uncommon in combinations problems...

_________________
Emily Sledge
Instructor
ManhattanGMAT