A pod of 6 dolphins - Word translation Chap 4 Q 5

[mkgeetha]

Hi ,
The question is :
A pod of 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 ?

Before I read the chapter and applied the 'anagram' strategy, I rightly calculated the answer as 3! x 3!.
But afer reading the chapter I am confused as to why the answer cannot be 6! / (3! x3!) .

Isn't this similar to MMMFFF anagram ? Could someone please help ?

Thanks
Geetha

[sanjeev]

Hi Geetha,

I haven't looked in the chapter as such. But i think its not an MMMFFF anagram. Instead , it is FFF
MMM, because question clears says that female have to be in front and male to be in rear.

This means you can arrange 3 females in 3! ways and 3 males in 3! ways. Hence the answere 3! * 3!.


Thanks
Sanjeev

[sanjeev]

Hi Geetha,

I haven't looked in the chapter as such. But i think its not an MMMFFF anagram. Instead , it is
FFF
MMM
because question clears says that female have to be in front and male to be in rear.

This means you can arrange 3 females in 3! ways and 3 males in 3! ways. Hence the answere 3! * 3!.


Thanks
Sanjeev[/quote]

[CD]

Geetha,

I'm not sure whether Sanjeev was joking with his answer, but I assume it didn't answer your question.

I am not an expert, but here's the way I think about it:

Putting a 3! * 3! in the denominator means that it doesn't matter which male is in each specific position, or which female is in each specific position, when it fact that's exactly what we need to focus on.

Let me put it another way. Another example from the strategy guide is something like the following: "A person is putting together a box of chocolates as a gift. The box has four compartments lined up next to eachother. There are two almond chocolates and two caramel chocolates. How many different ways can the chocolates be arranged in the compartments?" The answer to this would be 4! / (2! * 2!), because there is no difference between one almond chocolate and the other. A1 A2 C1 C2 is the same as A2 A1 C1 C2.

In the case of the dolphins, the lineup of M1 M2 M3 F1 F2 F3 is distinct from M2 M1 M3 F1 F2 F3. Because order here matters, it doesn't make sense to put any factorials in the denominator. The reason the answer isn't 6! is because there is a restriction on the arrangement. After one gender has taken up the first three spots, on the last three spots are available. Hence 3! * 3!.

[rfernandez]

I think everyone is in agreement here, and that Sanjeev wasn't joking. The problem is set up in such a way that you have to treat the males separately from the females.

The males can be arranged in 3! ways. The females can be arranged in 3! ways. Then, combine these two separate sub-problems by multiplying the two sets of arrangements: 3! * 3! = 36.

Regarding the anagram method, there's no way that I can think of to use that method with all six dolphins at once. This is because that method would assume that any dolphin can assume any position, which the problem stipulates is not possible.

Rey

[Guest]

Thanks, everybody. I think I understand it now. This is an 'arrangement' question where the order matters. So I cannot apply anagram strategy.

Regards,
Geetha

[rfernandez]

Glad it was helpful!

[dk]

How does 3! * 3! ensure that the females are at the front of the single line formation?

Also, when order matters, is it best to use the slot method instead of the anagram method?

[jnelson0612]

How does 3! * 3! ensure that the females are at the front of the single line formation?

Also, when order matters, is it best to use the slot method instead of the anagram method?

By setting up your equation of 3! * 3! you are specifying that the females are at the front of the line. Here's how:

Let's assume that your female dolphins are A, B, and C
Your male dolphins are D, E, and F

In slot one, we need a female dolphin. How many options are there? 3
In slot two, we need a female dolphin. How many options remain (since one dolphin was already chosen for slot one)? 2
In slot three, we need a female dolphin. How many options remain? 1

Thus, the females can be arranged in 3*2*1 ways. Then we go through the same process for the male dolphins in slots four, five and six, and again we have 3*2*1.

Multiplying the two arrangements together, we get 3!*3!.

As for slot vs. anagram, either of those can work when order matters. The slot is often just a mathematically simplified version of the anagram. Let me know if you need further explanation here.

[Nz]

Do we assume that the females are unique because it's not mentioned in the question that they are identical ?? I understand from the chapter that if they are identical then we divide by that factorial which is what I was doing.

I'm also not clear how the order matters here. If they are unique, then order matters, right?

Thanks!
Confused

[tim]

with unique things, order will generally matter. usually, if something can be mass produced in a factory, multiple copies are considered indistinguishable. however, you will probably offend a female dolphin by telling her that you think she is just the same as all other female dolphins. :) let common sense be your guide here..

[adm45]

Let me put it another way. Another example from the strategy guide is something like the following: "A person is putting together a box of chocolates as a gift. The box has four compartments lined up next to eachother. There are two almond chocolates and two caramel chocolates. How many different ways can the chocolates be arranged in the compartments?" The answer to this would be 4! / (2! * 2!), because there is no difference between one almond chocolate and the other. A1 A2 C1 C2 is the same as A2 A1 C1 C2.



Why can't you use the slot method? 4! equals 24.
If there are 4 compartments and each chocolate is indistinguishable then the slots are numbered 4 3 2 1.

Thanks

[jlucero]

adm45, I'm not sure if you had a question here, but I do want to point out the difference between the two problems. In the dolphin problem, switching two females is a different arrangement, while in yours, switching the almond chocolates is insignificant. In the original, treat the females and males as separate b/c they need to be in a certain order. But in yours, the chocolates can go anywhere, HOWEVER, we need to divide by 2!*2! because switching the same chocolates doesn't matter.