+91 Four squares are joined together to form an L-shaped piece. In how many ways can you place the L-shaped piece on a 5 x 5 grid? The L-shaped piece can be rotated and/or reflected.
I started counting normally just fitting it wherever but I don't think this is efficient. So I started finding all the cases that won't work but this is taking to long. Can anybody help me?
Is there a quicker way to count this?
-cosign
+2668 The 4 squares are unit squares that form an L shape that is 3 high and 2 wide right?
+2668
There are \(4\) ways to arrange the L so the bottom \(2\) peices are in the bottom row.
There are \(3\) rows that you can do this on, so there are \(12\) ways to arrange the L this way.
There are \(3\) additional ways to rotate them \(90^\circ\) with the same concept. This means that there are a total of \(48\) ways with rotations.
There are \(3\) ways to reflect the L , with each one having \(12\) ways to arrange them.
This makes for a total of \(\color{brown}\boxed {84}\) ways.
+2668 _ _ _ _ _
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X
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X
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X X
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Each X represents where one block would be. The other possible combinations would involve transforming the L \(1-3\) units to the right.
