Problem of distributing elements with restriction

[Nube]

Hello.
Six dogs and two cats have four hiding places to shelter from the rain. How can they
distribute the eight animals in the four hiding places, knowing that all the hiding places will be used and
also can not have dogs or cats in the same hiding place?

I can't say it in a way that I think will work, reading it comes to mind counting functions, the inclusion-exclusion principle, or dealing the red balls first and then the blue ones and using the product rule, but i don't think it works like that. So when I say the principle of inclusion-exclusion, it's because I think that having all the ways to fix all 8 balls, regardless of whether they stay in the same box, then the ones that are together could be eliminated, but that's the problem.

 

[Deleted member 4993]

Hello.
Six dogs and two cats have four hiding places to shelter from the rain. How can they
distribute the eight animals in the four hiding places, knowing that all the hiding places will be used and
also can not have dogs or cats in the same hiding place?

I can't say it in a way that I think will work, reading it comes to mind counting functions, the inclusion-exclusion principle, or dealing the red balls first and then the blue ones and using the product rule, but i don't think it works like that. So when I say the principle of inclusion-exclusion, it's because I think that having all the ways to fix all 8 balls, regardless of whether they stay in the same box, then the ones that are together could be eliminated, but that's the problem.

First pair-off the cats - you can do that only one way.

How many ways can you par-off the six dogs?

Now how many ways can you place these four pairs in in four places?

 

[Nube]

There I got confused writing, because I tried to consider dogs as little red balls and cats as little blue balls.

 

[Nube]

First pair-off the cats - you can do that only one way.

How many ways can you par-off the six dogs?

Now how many ways can you place these four pairs in in four places?

Why can the cats be distributed in only one way? Wouldn't they be combinations with a 6 in 2 repetition?

 

[Deleted member 4993]

Why can the cats be distributed in only one way? Wouldn't they be combinations with a 6 in 2 repetition?

Not distributed - but paired - there is only one pair of cat.

 

[Steven G]

IF you paired two cats, then there is only one way to pair them. Namely they both get paired.

 

[Dr.Peterson]

Hello.
Six dogs and two cats have four hiding places to shelter from the rain. How can they
distribute the eight animals in the four hiding places, knowing that all the hiding places will be used and
also can not have dogs or cats in the same hiding place?

I can't say it in a way that I think will work, reading it comes to mind counting functions, the inclusion-exclusion principle, or dealing the red balls first and then the blue ones and using the product rule, but i don't think it works like that. So when I say the principle of inclusion-exclusion, it's because I think that having all the ways to fix all 8 balls, regardless of whether they stay in the same box, then the ones that are together could be eliminated, but that's the problem.

Are we supposed to consider the animals as distinguishable (e.g. each has its own name), or indistinguishable (which would be true if you equate them just to balls of two colors)? I'll assume the former; I'll call the dogs A, B, C, D, E, and F, and the cats 1 and 2.

I'll also assume the hiding places are distinct, represented by _ _ _ _ in a fixed order.

The next thing I'd do is to write out some sample arrangements, to get a feel for what they look like. For example, if I am right, we could have

ABCDE F 1 2​

12 AB CD EF​

BDE 2 1 AC​

2 BDE 1 AC​

Now I'd think about how to count them. My first thought is to start with two cases: cats together, cats apart.

Note that I didn't start by picking a tool; I started by looking carefully at the job to be done.

 

[Nube]

So there are C(6,2) ways to pair dogs?

 

[Nube]

Are we supposed to consider the animals as distinguishable (e.g. each has its own name), or indistinguishable (which would be true if you equate them just to balls of two colors)? I'll assume the former; I'll call the dogs A, B, C, D, E, and F, and the cats 1 and 2.

I'll also assume the hiding places are distinct, represented by _ _ _ _ in a fixed order.

The next thing I'd do is to write out some sample arrangements, to get a feel for what they look like. For example, if I am right, we could have

ABCDE F 1 2​

12 AB CD EF​

BDE 2 1 AC​

2 BDE 1 AC​

Now I'd think about how to count them. My first thought is to start with two cases: cats together, cats apart.

Note that I didn't start by picking a tool; I started by looking carefully at the job to be done.

I think they are indistinguishable, everything I put is the only information I have. I talked about the inclusion-exclusion principle because just this exercise was in a list of this topic.

 

[Deleted member 4993]

I had wrongly assumed that you can only hide two animals in one place. If you can hide more than two animals (dogs in this case) then of course you need to consider the groupings suggested by @Dr.Peterson .

 

[Dr.Peterson]

I think they are indistinguishable, everything I put is the only information I have. I talked about the inclusion-exclusion principle because just this exercise was in a list of this topic.

Please show us the exercise exactly as given to you, including possibly other exercises in the set, the instructions, and details about the section it is in and what methods you have learned. This does not strike me as a problem in which inclusion-exclusion would be useful, though that depends on how you approach a problem.

In particular, I need some reason to take the animals and/or the places as indistinguishable. All dogs and cats I know have names.

If we do take the animals as indistinguishable, but the places as distinguishable, as I think you are taking it, then example arrangements might be

DDDDD D C C​

CC DD DD DD​

DDD C C DDD​

C DDDD C DD​

I would still be inclined to consider two cases, according to whether the cats are together or not. And I would not be surprised to use "stars and bars", though it may not be needed.

 

[Nube]

Please show us the exercise exactly as given to you, including possibly other exercises in the set, the instructions, and details about the section it is in and what methods you have learned. This does not strike me as a problem in which inclusion-exclusion would be useful, though that depends on how you approach a problem.

In particular, I need some reason to take the animals and/or the places as indistinguishable. All dogs and cats I know have names.

If we do take the animals as indistinguishable, but the places as distinguishable, as I think you are taking it, then example arrangements might be

DDDDD D C C​

CC DD DD DD​

DDD C C DDD​

C DDDD C DD​

I would still be inclined to consider two cases, according to whether the cats are together or not. And I would not be surprised to use "stars and bars", though it may not be needed.

is the literal letter of the exercise, I don't think it is useful to use stars and bars as I could not be sure that there were separate cats and dogs.

 

[Dr.Peterson]

is the literal letter of the exercise, I don't think it is useful to use stars and bars as I could not be sure that there were separate cats and dogs.

Do whatever you like; there are many ways to do these. Just show us some more work that we can comment on. I've said all I want to say without additional feedback.

 

[Steven G]

I had wrongly assumed that you can only hide two animals in one place. If you can hide more than two animals (dogs in this case) then of course you need to consider the groupings suggested by @Dr.Peterson .

You need to go to the dog house in the corner for that error. You should stay there for 1 cat year.

 

[Nube]

Do whatever you like; there are many ways to do these. Just show us some more work that we can comment on. I've said all I want to say without additional feedback.

I did typical exercises and 90% of them were counting positive integer solutions with constraints and by the inclusion-exclusion principle removing them but nothing relevant to solve this.

 

[pka]

Six dogs and two cats have four hiding places to shelter from the rain. How can they
distribute the eight animals in the four hiding places, knowing that all the hiding places will be used and
also can not have dogs or cats in the same hiding place?

It appears that there are really only those two restriction given for sure.
It seems to this old counter that it is reasonable to assume that the hiding places are distinct.
But are the dogs indistinguishable or not? Cats?
Can two cats be in different hiding places? If so that leaves only two places for six dogs.
@Nube, it appears that you have many details to clear-up.

 

[Nube]

Parece que en realidad solo se dan esas dos restricciones con seguridad.
A este viejo contador le parece razonable suponer que los escondites son distintos.
¿Pero los perros son indistinguibles o no? ¿gatos?
¿Pueden dos gatos estar en diferentes escondites? Si es así, eso deja solo dos lugares para seis perros.
@Nube, parece que tiene muchos detalles que aclarar.

I would like to know it too, that is all the information of the exercise, if you hurry me I would think about it as I said above 2 blue and 6 red balls where they cannot be in the same box

 

[Dr.Peterson]

is the literal letter of the exercise

I'd missed this, which pka bolded:

Six dogs and two cats have four hiding places to shelter from the rain. How can they
distribute the eight animals in the four hiding places, knowing that all the hiding places will be used and
also can not have dogs or cats in the same hiding place?

I read it as "dogs and cats", meaning each place can hold either one or more cats, or one or more dogs, but not some of each.

But if it means exactly what it says, and you copied it exactly, then you can't have more than one dog in the same place, or more than one cat in the same place. Since there are more dogs than places, the answer therefore is zero. ;-)

I am hoping you didn't really copy it correctly as you claim! [Or, see below, that you didn't translate it correctly]

I did typical exercises and 90% of them were counting positive integer solutions with constraints and by the inclusion-exclusion principle removing them but nothing relevant to solve this.

Can't you show some examples of this, which is rather broad? And some examples given in the text?

I would like to know it too, that is all the information of the exercise, if you hurry me I would think about it as I said above 2 blue and 6 red balls where they cannot be in the same box

I see that you wrote this in Spanish; that is not necessarily a reason not to show us the actual original of the problem, or others!

As I've said, although the problem as you quoted it doesn't clarify what is distinguishable, the instructions or previous examples may well give clues. Lacking that, we can only assume your interpretation is correct, which is not a problem.

But we do need to see actual work from you that we can then modify. Just try something like what I suggested. It is not impossible, though it may not use exactly the methods you are currently learning. And if you happen to have been given the "correct" answer, that would help with the interpretation.

 

[Nube]

I'd missed this, which pka bolded:

I read it as "dogs and cats", meaning each place can hold either one or more cats, or one or more dogs, but not some of each.

But if it means exactly what it says, and you copied it exactly, then you can't have more than one dog in the same place, or more than one cat in the same place. Since there are more dogs than places, the answer therefore is zero. ;-)

I am hoping you didn't really copy it correctly as you claim! [Or, see below, that you didn't translate it correctly]

Can't you show some examples of this, which is rather broad? And some examples given in the text?


I see that you wrote this in Spanish; that is not necessarily a reason not to show us the actual original of the problem, or others!

As I've said, although the problem as you quoted it doesn't clarify what is distinguishable, the instructions or previous examples may well give clues. Lacking that, we can only assume your interpretation is correct, which is not a problem.

But we do need to see actual work from you that we can then modify. Just try something like what I suggested. It is not impossible, though it may not use exactly the methods you are currently learning. And if you happen to have been given the "correct" answer, that would help with the interpretation.

In each house, there can be any number of dogs or cats, the difference is that there cannot be dogs and cats TOGETHER.

 

[Dr.Peterson]

In each house, there can be any number of dogs or cats, the difference is that there cannot be dogs and cats TOGETHER.

I'd still like to see an image of the actual problem, but I'm willing to stipulate that your interpretation is correct.

The method I suggested, using two cases (cats together, cats separate) gives a quick an easy answer; but since the context suggests that you are expected to use inclusion-exclusion, you need to find a way to express the problem so that it can be applied; that is, you want to see it as counting something such that none of several conditions are true.

Here is one way I can imagine, though I haven't pursued it all the way:

Take my examples, and look at them in a way similar to stars-and-bars:

DDDDD|D|C|C​

CC|DD|DD|DD​

DDD|C|C|DDD​

C|DDDD|C|DD​

Here are some excluded arrangements:

DDDD|D|CD|C​

C|DD|DC|DDD​

CC||DDDD|DD​

CC|DDDD|DD|​

Using this representation, you want to arrange the symbols DDDDDDCC||| such that there is no CD, DC, ||, or | at either end (no C and D in the same place, and no empty place). Try applying the rule to that. I think it will be far harder than my way, but should give you the practice you are expected to get. There may well be a better way.