A 1×6 square unit rectangular grid has to be covered with six 1×1 square unit tiles. Three of the tiles are yellow, two are white, and one is blue. In how many ways can this task be done if
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at least one pair of neighboring tiles are the same color?
To figure out how many ways the rectangular grid can be covered, we can use a counting method.
First, let's consider the case where there are no restrictions on the tile colors. In this case, we have a total of 6 tiles to cover the 6 units in the grid. We can think of this as placing 6 distinguishable objects (the tiles) into 6 distinguishable units in the grid. The number of ways to do this is simply 6!
Next, let's find the total number of ways to cover the grid without any neighboring tiles having the same color. We need to count the number of color arrangements that violate this condition and subtract it from the total number of ways to cover the grid.
There are three possible cases where neighboring tiles have the same color:
1. When the leftmost and the second leftmost tiles have the same color.
2. When the middle two tiles have the same color.
3. When the rightmost and the second rightmost tiles have the same color.
Let's analyze each case separately:
1. When the leftmost and the second leftmost tiles have the same color:
We have two possibilities:
- Yellow, Yellow, White, Blue, White
- Yellow, Yellow, Blue, White, White
For each of these, the remaining two colors can be arranged in 2! ways (white, blue or blue, white). So there are a total of 2 x 2! = 4 ways in this case.
2. When the middle two tiles have the same color:
We have three possibilities:
- Yellow, White, White, Blue, Yellow
- Yellow, Blue, White, White, Yellow
- Blue, White, White, Yellow, Yellow
For each of these, the remaining two colors can be arranged in 2! ways. So there are a total of 3 x 2! = 6 ways in this case.
3. When the rightmost and the second rightmost tiles have the same color:
We have two possibilities:
- Yellow, White, Blue, White, Yellow
- Yellow, White, White, Blue, Yellow
For each of these, the remaining two colors can be arranged in 2! ways. So there are a total of 2 x 2! = 4 ways in this case.
Adding up the possibilities from each case, we get a total of 4 + 6 + 4 = 14 ways to cover the grid without any neighboring tiles having the same color.
Therefore, to find the final answer, we subtract the number of ways without the restriction from the total number of ways to cover the grid:
Total ways = 6! - 14 = 720 - 14 = 706 ways.
So there are 706 ways to cover the 1×6 square unit rectangular grid with six 1×1 square unit tiles, where at least one pair of neighboring tiles are the same color.