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  • In a 4x4 grid, each cell is colored either red, blue, or green. How many different ways can you color the grid such that each row and column contains all three colors, no two adjacent cells have th

In a 4x4 grid, each cell is colored either red, blue, or green. How many different ways can you color the grid such that each row and column contains all three colors, no two adjacent cells have the same color, and no two rows or columns are identical?

Option: 1

10,200


Option: 2

7,200


Option: 3

3,072


Option: 4

8,460


Answers (1)

best_answer

To solve this problem, we can break it down into smaller steps.

Step 1: Determine the number of ways to color the first row.

In the first row, we have 3 options for the first cell (red, blue, or green). For the second cell, we have 2 options (we cannot choose the same color as the first cell). Similarly, we have 2 options for the third cell and 2 options for the fourth cell.

The number of ways to color the first row is: 3 \times 2 \times 2 \times 2=24.

Step 2: Determine the number of ways to color the second row.

In the second row, we have 2 remaining colors to choose from (since we have already used one color in the first row). For each cell, we have 2 options (we cannot choose the same color as the corresponding cell in the first row).
The number of ways to color the second row is:2\times 2 \times 2 \times 2 = 16.

Step 3: Determine the number of ways to color the third row.

In the third row, we have 1 remaining color to choose from (since we have already used two colors in the first two rows). For each cell, we have 2 options (we cannot choose the same color as the corresponding cell in the first two rows).
The number of ways to color the third row is:1\times 2 \times 2 \times 2 = 8.

Step 4: Determine the number of ways to color the fourth row.

In the fourth row, we have no remaining color choices (since we have already used all three colors in the previous rows). For each cell, we have 1 option.

The number of ways to color the fourth row is: 1\times 1 \times 1 \times 1 = 1.
 

Step 5: Determine the total number of ways to color the grid.

To find the total number of ways to color the grid, we multiply the number of ways to color each row together.

Total number of ways = (number of ways to color the first row) \times (number of ways to color the second row) \times(number of ways to color the third row) \times (number of ways to color the fourth row)

Total number of ways = 24 \times 16\times 8\times 1 = 3,072.

Therefore, there are 3,072 different ways to color the 4x4 grid such that each row and column contains all three colors, no two adjacent cells have the same color, and no two rows or columns are

 

Posted by

Kuldeep Maurya

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