Combinatorics

[happyface101]

Hi, I would really appreciate it if you can help me with this problem from the Number Properties book page 53.

A pod of 6 dolphins always swims single file, with 3 females at the front and 3 males in the rear. In how many different arrangements can the dolphins swim?

I'm having trouble with (1) seeing why 3!*3! is the answer, and (2) how I can solve this problem with the method of x! / (y! chosen * z! not chosen), since it seems like all the other problems in this section can be solved that way. thanks so much.

[RonPurewal]

(1) seeing why 3!*3! is the answer

there are 3! ways to arrange the females, and 3! ways to arrange the males. (this is, in my opinion, the only thing in combinatorics that's actually worth memorizing: there are n! ways to order n different things.)

[RonPurewal]

also, more importantly, if you don't IMMEDIATELY make this connection, JUST MAKE LISTS.
if the female dolphins are "1", "2", and "3", then the possible orders are:
123
132
213
231
312
321
...well, that's not very hard (nor is it time-consuming... 10 seconds, maybe). 6 orders (= same as 3!).

there are the same number of males, so a fortiori there will be the same number of ways to arrange the males. (if you blank out on that, just take 10 more seconds and make another list.)

the two arrangements don't depend on each other, so you can multiply them: 6 x 6 = 36.

--

you should NOT pooh-pooh this sort of approach, by the way. nearly ALL of GMAC's combinatorics problems can be solved by making lists—and, moreover, the only ones that can't are, almost invariably, the most basic ones.

when i'm personally faced with a gmat combinatorics problem, my hierarchy of methods is...
plan A: make a list and count the things in it
plan B: everything else.

[RonPurewal]

(2) how I can solve this problem with the method of x! / (y! chosen * z! not chosen)

you can't, because there is no "chosen"/"not chosen". you're putting everyone into the orderings; it's the No Dolphin Left Behind program.

[FredericB588]

I would like to re-open this question please as I don't completely understand why this problem (problem 3 of chapter 4 of strategy guide 5, 6th edition) why the chosen/not chosen approach does not work.

Specifically:

If I take the 3 female dolphins to start with and I pick one female dolphin to lead the pod, why are the other two female dolphins "not chosen" to be the lead? I would have thought that they would not be chosen and therefore that I could use the anagram method to solve the problem in a similar fashion as I would use the anagram method to solve the chocolate box problem (problem 2 of chapter 4 of strategy guide 5, 6th edition).

Thank you.

Frederic

[Sage Pearce-Higgins]

The dolphin problem is a good one to get a conceptual handle on what it means to count combinations of things. In my opinion, this kind of understanding is what really counts on GMAT. Methods such as the anagram method are useful shortcuts, but can get you into trouble unless you understand them.

If you have 3 objects, then counting the number of ways that they can be ordered can be done by thinking 'Hey, let's break this down into imaginary choices. For the first object, I have 3 choices, i.e. I can choose any of the 3 objects. This leaves 2 objects left, so I have 2 choices for the second object. Clearly, by the time I've got to the third object, there's only 1 object to choose. Translating that into Math makes 3 * 2 * 1.' Now, this doesn't mean that we've actually chosen one object to be first. Actually, this is the bit that makes everyone's head hurt. We've though of the 3 possible choices that we could make. We haven't actually picked out an individual example.

I suggest that you give the objects names, such as A B C, and then write out the possible orders that you could put them in as an exercise to help yourself understand how this confusing idea of 'imaginary choices' works with a real example.