Combinations/Permutations of tile placement

[j0hn_]

I have a grid and a set of tiles, and need to find the number of permutations for placing said tiles.

If I have a 3x3 grid and 9 tiles to place, then P = 9!
With 12 tiles P = 12!/(12-9)!
I think I've got that basic idea correct, right?

I'm getting lost as I change the rules...

If a space can be empty, does P =n!/(n-r)! + r^2 (r^2 because that's the number of ways tiles can be empty or full?)
What If a space can contain only one specific color (or be blank)?

And the final complication-- Given tiles of different colors; 4 blue, 4 red, 4 green, 4 yellow, if we only care about color, is this now a combination, not permutation? How is that calculated?


Appreciate any help. Been frying my brain and drawing little grids with filled in squares all over the place, it's been a long time since I thought about this stuff...

 

[Deleted member 4993]

I have a grid and a set of tiles, and need to find the number of permutations for placing said tiles.

If I have a 3x3 grid and 9 tiles to place, then P = 9!
With 12 tiles P = 12!/(12-9)!
I think I've got that basic idea correct, right?

I'm getting lost as I change the rules...

If a space can be empty, does P =n!/(n-r)! + r^2 (r^2 because that's the number of ways tiles can be empty or full?)
What If a space can contain only one specific color (or be blank)?

And the final complication-- Given tiles of different colors; 4 blue, 4 red, 4 green, 4 yellow, if we only care about color, is this now a combination, not permutation? How is that calculated?


Appreciate any help. Been frying my brain and drawing little grids with filled in squares all over the place, it's been a long time since I thought about this stuff...

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem.

 

[Steven G]

In a 3x3 grid there are spaces for 9 tiles to be placed. I am assuming that you can put at most tile in a space.
Now you have 12 tiles and want to know in how many ways you can place these 12 tiles in the 9 spaces? I would think that this answer is obvious.

 

[j0hn_]

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem.

"Show your beginning work, or ask a specific question about the exercise, or explain why you're stuck."

I asked specific question, you might want to remove the ORs if you demand all three parameters met. But never mind, will seek help elsewhere.
What a helpful and welcoming community you have here!

 

[Dr.Peterson]

I have a grid and a set of tiles, and need to find the number of permutations for placing said tiles.

If I have a 3x3 grid and 9 tiles to place, then P = 9!
With 12 tiles P = 12!/(12-9)!
I think I've got that basic idea correct, right?

I'm getting lost as I change the rules...

If a space can be empty, does P =n!/(n-r)! + r^2 (r^2 because that's the number of ways tiles can be empty or full?)
What If a space can contain only one specific color (or be blank)?

And the final complication-- Given tiles of different colors; 4 blue, 4 red, 4 green, 4 yellow, if we only care about color, is this now a combination, not permutation? How is that calculated?


Appreciate any help. Been frying my brain and drawing little grids with filled in squares all over the place, it's been a long time since I thought about this stuff...

Part of the trouble is that you have to very carefully state exactly what you want in this sort of problem, and you haven't. Combinatorics is notoriously easy to misstate, misread, or mis-think!

If you want to place 9 distinct tiles in 9 distinct places, and order matters, then that is a permutation of the entire set, 9P9 = 9! = 362,880.

If you want to place some subset of 12 distinct tiles in 9 distinct places, and order matters, then that is a permutation of a subset, 12P9 = 9!/3! = 79,833,600, as you say.

If you want to place a subset of n tiles in a subset of r places (that is, you can leave any number of the places empty), then you have a much harder problem, and there may not even be a simple formula -- I haven't yet tried to work it out. Your attempt, however, is certainly wrong. As you modify a problem, you can change the necessary work drastically, and should not expect it to be easy. (I commonly don't trust my answer until I can find two different ways to think about it that give the same result.)

Your last example is even harder, I imagine, and is less clear. You evidently care only about color, so the tiles are not distinct; but that doesn't make this a combination problem. Are they still going into a grid with 9 spaces? Or are you willing to start with the easier problem with 12 spaces? That's the one I'd like to interact with you on.

In any case, the main help I can give at this point is to tell you not to expect these all to be easy.

"Show your beginning work, or ask a specific question about the exercise, or explain why you're stuck."

I asked specific question, you might want to remove the ORs if you demand all three parameters met. But never mind, will seek help elsewhere.
What a helpful and welcoming community you have here!

What we mean by "specific question" is, a specific question about the problem. You've just tossed out a bunch of different problems at different levels, and shown work for the easy ones, without showing your thoughts about the interesting ones.

And the answer we give to a specific question will often be a further question. We want to see what sort of help will be of use to you.

But you seem to be giving up before you've really interacted with us. So I guess I've wasted my effort to do so.