In how many patterns can six identical L-shaped pieces, each consisting of three unit squares, be placed to perfectly cover a fixed 3 by 6 board?

Question

In how many patterns can six identical L-shaped pieces, each consisting of three unit squares, be placed to perfectly cover a fixed 3 by 6 board?

I know this problem is dealing with permutations, any hints are greatly appreciated.

Answer 1

I wanted to add this as a comment, but I need to have at least 50 reputations to add a comment. Therefore, I'm adding this as an answer. Even though I'm not sure whether it is correct or not.

So, you have to put $3$ L shaped pieces in a row to cover the rectangle. Since the 3 pieces are identical, it doesn't matter which is which but the thing that matters is the shape of the L that you use. For example, starting from the bottom row, you have $2$ choices to fill the first $4$ blocks: |_ or _| (draw a figure). Once you have made that choice, the upper piece above it will be determined automatically. Therefore, you have $2^3$ choices to put $3$ L-shapes in the bottom row to cover this rectangle and that's it.

Answer 2

Consider the top left unit square. There are three different ways an L-shaped piece can cover that square:

For the first two cases, there is only one way to place another piece to cover the lower left corner. In the last case, there is no way to place another piece to cover the lower left corner without overlapping the first piece. In both of the first two cases, the two leftmost columns will be covered. So, we can use this logic again, on the top left square which has not yet been covered. We have two choices of how to cover the first two columns, two choices of how to cover the next two columns, and two choices of how to cover the last two columns, so there are $2 \cdot 2 \cdot 2=8$ total ways to cover the entire board.