CAT #3, Problem #14 - "Goodbuddies"

[tammylhan]

I felt that the explanation for this problem was a bit lacking and it took me a while to really understand the combinatorics strategy that we learned in our Word Translations Strategy Guide - Combinatorics Chapter.

The explanation reads:
"Ignoring Frankie's requirement for a moment, observe that the six mobsters can be arranged 6! or 6 x 5 x 4 x 3 x 2 x 1 = 720 different ways in the concession stand line. In each of those 720 arrangements, Frankie must be either ahead of or behind Joey. Logically, since the combinations favor neither Frankie nor Joey, each would be behind the other in precisely half of the arrangements. Therefore, in order to satisfy Frankie's requirement, the six mobsters could be arranged in 720/2 = 360 different ways."

For someone who is not great at math, "logically, the combinations not favoring a person" are not clear in an algebraic manner. I am certainly not a qualified MGMAT instructor/representative, so if anyone with MGMAT authority sees this, and feels I have used improper logic, please correct me if I misstate anything.

In this problem, Joey would only be able to stand in line in positions 1 through 5, of 6 (he cannot stand in 6, or Frankie cannot stand behind him). If Joey were to occupy the first spot in line, there are 5 spots behind him that Frankie may elect to stand in (as Frankie must stand behind him, but not necessarily directly behind him), and the other 4 mobsters can stand in any order they want (use 4!), so you get 5*4! = 120. If Joey occupies the second spot in line, Frankie can now only select from 4 spots to stand behind him (because he cannot stand in front of him), and the other 4 mobsters still have freedom of choice, (use 4!), so you get 4*4! = 96. Thus, when Joey takes up spot 3, Frankie now has 3 places to choose from and you get 3*4! = 72. When Joey gets to the 4th spot in line, Frankie now only has 2 options, and you get 2*4! = 48. Lastly, if Joey elects to stand in the 5th spot in line, Frankie only has one option - the 6th spot, so you get 1*4! = 24. Adding up all these different scenarios, you get 120 + 96 + 72 + 48 + 24 = 360 different scenarios that Frankie will satisfy his requirement of being behind Joey.

Alternatively, I also felt MGMAT's strategy/my instructor's "Glue Method" suggestion for combinatorics problems with constraints is somewhat helpful in understanding this problem. The only reason it may be a little misleading is because the question stem specifies that Frankie must stand behind Joey, BUT NOT NECESSARILY BEHIND HIM. If they travel together as JF, (but not FJ, because Joey must PRECEDE Frankie in line), there are now effectively 5 "people". There are 5 scenarios, similar to those mentioned in the previous paragraph, but this time they are stuck together. In the first scenario, JF has 5 options of where they choose to stand, and to make things easier, why not the first spot in line, and the other 4 mobsters have freedom in arrangement (we use 4!) and the first scenario works with 5*4!, then in the second through fifth scenario, refer to above paragraph and add up the totals, giving you the 360 arrangements that satisfy the constraint.

[jnelson0612]

Hi Tammy,
I'm sorry the explanation wasn't more straightforward to you. It sounds as if you grasp the problem now, but let's review the "official" explanation one more time to see if it can help you.

First we determine the total number of possibilities.
First spot= 6 possibilities (any of the six could stand here)
Second spot = 5 possibilities (since one person has already been selected for the first spot)
Third spot= 4 possibilities
and so on.

Thus, we have 6! possible arrangements of mobsters, or 720.

We know that Frankie wants to always keep Joey ahead of him. In how many arrangements will that be the case? Well, they can never stand next to each other, so Joey is either ahead of or behind Frankie. It makes sense to assume that in half of the arrangements Joey will be ahead of Frankie and in half Frankie will be ahead of Joey. Thus, half of 720 is 360.

I hope this helps clear our explanation just a little bit. You did nice work in figuring things out on your own.

[brooks.t.slater]

Just thought I would clarify this comment -- "It makes sense to assume that in half of the arrangements Joey will be ahead of Frankie and in half Frankie will be ahead of Joey." -- Look at this comment in terms of probability:

POSITION THAT JOEY STANDS IN - PROBABILITY THAT FRANKIE STANDS BEHIND JOEY

1 - 100%
2 - 4/5
3 - 3/5
4 - 2/5
5 - 1/5
6 - 0%

If you multiply each of these probabilities by 1/6 and sum the result (refer to review material discussing complex probability), you will see that there is a 50% probability that Joey stands in front of Frankie.

[jnelson0612]

Just thought I would clarify this comment -- "It makes sense to assume that in half of the arrangements Joey will be ahead of Frankie and in half Frankie will be ahead of Joey." -- Look at this comment in terms of probability:

POSITION THAT JOEY STANDS IN - PROBABILITY THAT FRANKIE STANDS BEHIND JOEY

1 - 100%
2 - 4/5
3 - 3/5
4 - 2/5
5 - 1/5
6 - 0%

If you multiply each of these probabilities by 1/6 and sum the result (refer to review material discussing complex probability), you will see that there is a 50% probability that Joey stands in front of Frankie.

Wow, very nice! Way to show the proof for this problem! :-)