You have a pile of 1 x 2 unit dominoes. There's one way to tile a 1x2 area, 2 ways to tile a 2x2 area, 3 ways to tile a 3x2 area, and 5 ways to tile a 4x2 area. How many ways are there to tile a 5x2 area with your 1x2 domino tiles?
I find 8.
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To determine the number of ways to tile a 5x2 area with 1x2 domino tiles, we can use a recursive approach.
Let's consider the last column of the 5x2 area. It can be tiled in three different ways:
1. If the last column is tiled with a single vertical domino (|), then the remaining 4x2 area to the left can be tiled in a number of ways.
2. If the last column is tiled with two horizontal dominos (==), then the remaining 3x2 area to the left can be tiled in a number of ways.
3. If the last column is tiled with a single horizontal domino (=) and a single vertical domino (|), then the remaining 3x2 area to the left can be tiled in a number of ways.
Using this information, we can establish a recursive relation to calculate the number of ways to tile a 5x2 area:
T(5) = T(4) + T(3) + T(3)
Now, let's break it down further:
T(4) represents the number of ways to tile a 4x2 area, which we know is 5 from the given information.
T(3) represents the number of ways to tile a 3x2 area, which we also know is 3 from the given information.
Substituting these values into the recursive relation:
T(5) = 5 + 3 + 3
T(5) = 11
Therefore, there are 11 ways to tile a 5x2 area with 1x2 domino tiles.