Number of ways to color a grid

[Viking1]

Hello,
Kindly, can anyone solve the following:

How many combinations:

8 colors, which each can be used twice in the following grid

1 3 5 7
2 4 6 8

where a color can only appear like this
1 with 5
3 with 7
2 with 6
4 with 8

So, non-valid combinations would be
1 with 2
1 with 3
2 with 3
2 with 4 etc. etc.

Thanks a lot!

 

[jvicens]

Hello,

Thank you for your question. To solve this problem, we need to first understand the constraints given. We have 8 colors and each color can be used twice. We also have a grid with 4 rows and 2 columns, where the first column can only have the numbers 1, 2, 3, and 4, while the second column can only have the numbers 5, 6, 7, and 8. Additionally, we have the constraint that each color can only appear in a specific pair, for example, 1 can only appear with 5 and 2 can only appear with 6. This means that we cannot have a combination like 1 with 2 or 3 with 6.

To solve this problem, we can use the fundamental principle of counting. This principle states that if we have m ways to do one task and n ways to do another task, then the total number of ways to do both tasks is m x n. In this case, we have 8 colors to choose from for the first pair (1 with 5) and 7 colors left to choose from for the second pair (3 with 7). This gives us a total of 8 x 7 = 56 combinations for the first two pairs.

For the third pair (2 with 6), we have 6 colors left to choose from and for the fourth pair (4 with 8), we have 5 colors left to choose from. This gives us a total of 6 x 5 = 30 combinations for the third and fourth pairs.

Therefore, the total number of combinations for this grid is 56 x 30 = 1680.

I hope this helps to solve your problem. Let me know if you have any further questions.