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
b
no two neighboring tiles are the same color?
Well, it seems like this task requires some creative tile placement! Since no two neighboring tiles can be the same color, let's explore the possibilities:
1. Let's start by placing the yellow tiles first. We have 3 yellow tiles to place in 6 spots. We can do this in 6 choose 3 ways, which is C(6,3) = 20 ways.
2. Now, let's figure out how to place the white tiles. We have 2 white tiles to place in the remaining 3 spots. We can do this in 3 choose 2 ways, which is C(3,2) = 3 ways.
3. Lastly, we have the blue tile. Since it's the only tile remaining, there is only 1 way to place it.
Now, we multiply the number of ways for each step to find the total number of possible arrangements:
Total = 20 * 3 * 1 = 60
So, there are 60 different ways to place the tiles on the 1x6 grid without any neighboring tiles having the same color.