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?
6
24
120
360
720
According to me the anser should be 120
Please let me know the flaw in my Logic....
Let Six Mobserts be M1,M2,M3,M4,F(Frank),J(Joey)
Now according to question Frank needs to stand behind Joe,So let Us take a postion where in Joe Stands at the atart of Queue
M1,M2,M3,M4,F,J---- Frank can stand behind Joe in 5 Ways
M1,M2,M3,F,J,M4-----Frank can stand behind Joe in 4 ways
Carrying this logic forward we can say that there are 5*4*3*2=120 Positions in which Frank can stand behind Joe.......
But according to Manhattan GMAt the answer is 720.....Please help me find flaw in my Logic....
GMAT Forum
5 postsPage 1 of 1
- Abhimanyu Sood
- RonPurewal
- Students
- Posts: 19744
- Joined: Tue Aug 14, 2007 8:23 am
to the original poster:
the flaw in your logic is that you're only considering the cases in which frankie stands <i>directly</i> behind johnny, whereas the problem is unequivocal in stating that frankie is allowed to be more than one place behind johnny.
here are all the possibilities:
J F _ _ _ _
J _ F _ _ _
J _ _ F _ _
J _ _ _ F _
J _ _ _ _ F
_ J F _ _ _
_ J _ F _ _
_ J _ _ F _
_ J _ _ _ F
_ _ J F _ _
_ _ J _ F _
_ _ J _ _ F
_ _ _ J F _
_ _ _ J _ F
_ _ _ _ J F
that's fifteen possibilities. each allows 4 x 3 x 2 x 1 = 24 sub-possibilities for the remaining four mobsters, whose order is not restricted, so we have a grand total of 15 x 24 = 360 possibilities.
------------
also note that the fastest way to solve this problem is to make the following realization:
for every arrangement with frankie behind johnny, there is an equivalent arrangement with frankie in front of johnny - just switch them. because of this 1-1 correspondence, we know immediately that 1/2 of the arrangements have frankie in front of johnny, and the other 1/2 have frankie behind johnny. therefore, 1/2 of 720, or 360, is the answer.
although this solution is nice, i actually prefer the type of solution above, for at least two reasons:
1) very few gmat problems are amenable to this sort of symmetry argument, and
2) more importantly, this is the sort of argument that requires considerable familiarity with combinatorial math even to formulate. many students don't even trust arguments such as this one ('wait, i know it's not <i>that</i> simple; i feel as though i'm guessing')
the flaw in your logic is that you're only considering the cases in which frankie stands <i>directly</i> behind johnny, whereas the problem is unequivocal in stating that frankie is allowed to be more than one place behind johnny.
here are all the possibilities:
J F _ _ _ _
J _ F _ _ _
J _ _ F _ _
J _ _ _ F _
J _ _ _ _ F
_ J F _ _ _
_ J _ F _ _
_ J _ _ F _
_ J _ _ _ F
_ _ J F _ _
_ _ J _ F _
_ _ J _ _ F
_ _ _ J F _
_ _ _ J _ F
_ _ _ _ J F
that's fifteen possibilities. each allows 4 x 3 x 2 x 1 = 24 sub-possibilities for the remaining four mobsters, whose order is not restricted, so we have a grand total of 15 x 24 = 360 possibilities.
------------
also note that the fastest way to solve this problem is to make the following realization:
for every arrangement with frankie behind johnny, there is an equivalent arrangement with frankie in front of johnny - just switch them. because of this 1-1 correspondence, we know immediately that 1/2 of the arrangements have frankie in front of johnny, and the other 1/2 have frankie behind johnny. therefore, 1/2 of 720, or 360, is the answer.
although this solution is nice, i actually prefer the type of solution above, for at least two reasons:
1) very few gmat problems are amenable to this sort of symmetry argument, and
2) more importantly, this is the sort of argument that requires considerable familiarity with combinatorial math even to formulate. many students don't even trust arguments such as this one ('wait, i know it's not <i>that</i> simple; i feel as though i'm guessing')
5 posts Page 1 of 1