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Compute the number of ways to tile a 3 × 5 rectangle with one 1 × 1 tile, one 1 × 2 tile, one 1 × 3 tile, one 1 × 4 tile, and one 1 × 5 tile. (The tiles can be rotated, and tilings that differ by rotation or reflection are considered distinct.)

There are 2 ways you can stack 1x5 tile at the farther ends, there are 2 ways to stack 1X4 in a any other row, and there are 4 four ways you can stack the rest to a total of 16 ways. There is another configuration in which you stack 1x5 in the middle and 4 ways to stack 1x4 and 1x1 in the other farther row and 2 ways to stack 1x3 and 1x2 in the farther rows to a total of 8 to a sum total of $\boxed{24}$ can someone verify the number

Satish Ramanathan
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1 Answers1

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You correctly counted the arrangements where all tiles lie along rows. (It would be a lot simpler though to just say that you can join the $1$ and $4$ into a block of $5$ in $2$ ways and likewise join the $2$ and $3$ in $2$ ways, and then there are $3!=6$ ways to choose the rows for the $3$ resulting blocks, for a total of $2\cdot2\cdot6=24$.)

But you overlooked that you can also put the $1\times2$ tile across rows, with $4$ and $1+3=4$ next to it. For this, the $5$ needs to be in one of the $2$ end rows, the $2$ needs to be in one of the $2$ end columns, and then you have $2$ choices for the remaining rows and $2$ choices for the order of $1$ and $3$, for a total of $2\cdot2\cdot2\cdot2=16$.

All in all, that makes $24+16=40$ tilings.

joriki
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