Combinations - how many?

[Viking]

Hello,
Looking for help to solve the following:

Imagine a box with 8 spaces as follows (seen from above)

1	3	5	7
2	4	6	8

Imagine you have 16 balls split on 8 colors, so there´s two balls with same color.

You need to fill the spaces in the box using 4 pairs of balls, with the following limitation:

Same colored balls must be filled in the same spaces.

1 with 5
2 with 6
3 with 7
4 with 8

How many combinations are there and what is the formula to calculate it?

Thanks a lot.

 

[pka]

Hello,
Looking for help to solve the following:

Imagine a box with 8 spaces as follows (seen from above)

1	3	5	7
2	4	6	8

Imagine you have 16 balls split on 8 colors, so there´s two balls with same color.
You need to fill the spaces in the box using 4 pairs of balls, with the following limitation:
Same colored balls must be filled in the same spaces.
1 with 5
2 with 6
3 with 7
4 with 8
How many combinations are there and what is the formula to calculate it?

I know that this question is crystal clear to whom ever wrote it. It is not to me.
So here is a guess as to what it means.
There are eight choices for a colour for cell 1. Once that choice is made cell 5 is fixed.
So we have permutation of 8 taken 4: \(\displaystyle \mathcal{P}^8_4=1680\) SEE HERE

Now I doubt this interpretation.

 

[Viking]

I know that this question is crystal clear to whom ever wrote it. It is not to me.
So here is a guess as to what it means.
There are eight choices for a colour for cell 1. Once that choice is made cell 5 is fixed.
So we have permutation of 8 taken 4: \(\displaystyle \mathcal{P}^8_4=1680\) SEE HERE

Now I doubt this interpretation.

Hi,
Thanks for answering.
An example of a correct combination:
Red-White-Red-White
Blue-Green-Blue-Green

False combination:
Red-White-White-Red
Blue-Green-Blue-Green

Is your solution consistent with this?

Thanks

 

[pka]

An example of a correct combination:
Red-White-Red-White
Blue-Green-Blue-Green
Is your solution consistent with this?

NO! It is not like anything that you wrote.
Just read this:
Same colored balls must be filled in the same spaces.
1 with 5
2 with 6
3 with 7
4 with 8
What the heck does that mean if it does not mean that 1 & 5 have the same colour?
That list gives four pairs of two of the same colour.
Do you speak English? It certainly does not appear to be the case.
Why don't you get someone to help you translate this into readable English?

 

[Dr.Peterson]

Imagine a box with 8 spaces as follows (seen from above)

1	3	5	7
2	4	6	8

Imagine you have 16 balls split on 8 colors, so there´s two balls with same color.

You need to fill the spaces in the box using 4 pairs of balls, with the following limitation:

Same colored balls must be filled in the same spaces.

1 with 5
2 with 6
3 with 7
4 with 8

How many combinations are there and what is the formula to calculate it?

I'm not sure whether it is primarily an English-language issue, or lack of clarity in the problem itself. Here is a better version of what I think this is saying, and some questions that are still open in my mind:

We have a box with 8 compartments, numbered 1 through 8.​

We have 16 balls, two each of 8 colors.​

We need to put balls in the compartments; each compartment must contain balls of one color, and compartments 1 and 5, 2 and 6, 3 and 7, and 4 and 8 must contain balls of the same color.​

How many ways can the balls be put in the compartments?​

Here are the questions:

• Is it implied that only four colors can be used, so that only 8 balls are used?
• Can any compartment contain more than one ball, or be empty? (If the first answer is yes, then I think this is no, and my wording "each compartment must contain balls of one color" is not what you intended; you only meant that certain pairs of compartments must have the same color, though that is not at all what it means to say "Same colored balls must be filled in the same spaces.")
• Are the balls distinguishable? (Based on your example, I think not; only color matters.)

I think your example does fit what pka said; you mean that 1 and 5 are red, 2 and 6 are blue, 3 and 7 are white, 4 and 8 are green.

If my interpretation is correct, then compartments 5 through 8 are determined by the others, and can be ignored in the count. There are 8 choices for color 1, 7 for 2, 6 for 3, and 5 for 4, for a total of 8*7*6*5 = 1680 arrangements of colors.

 

[Viking]

Thanks to both,
Probably more a question of explaining the problem consistently and without misinterpretation possible, rather than needing someone to translate for me...

Only four colors can be used in single interpretation.
Each compartment can have one color only.
Only color matters.
The interpretation that 5 to 8 is determined by 1 to 4 is correct.

Correct combination as replied to pka:

Red	White	Red	White
Blue	Green	Blue	Green

 

[ksdhart2]

Frankly I'm feeling a bit insulted right about now. As if it's not bad enough that the OP was just given, apropos of no effort on their part, a full solution, now they want us to also do the thinking for them too, and decide if that solution is correct! Whatever happened to wanting to learn, and trying to figure out not just the solution itself but why the solution is what it is? ?

I'm trying not to be jaded and believe there are students out there who actually want to learn and aren't just seeking a silver spoon handout, but it's hard when it seems like everywhere all I see is no-effort posts and fishing for answers...

Anyway, sorry for derailing the topic. I just needed to vent, I guess.

 

[Viking]

I believe I gave a kindly thanks for the inputs, including explanation, all of which are indeed appreciated.
Best regards