Covering of a $2\times 3$ rectangle using smaller rectangles We have a $2 \times 3$ rectangle, that we wish to cover using $1 \times 2$ rectangles and $1 \times 1$ squares.  How many possible ways are there to do so?

Using $0$ and $3$ rectangles, it appears that there are only $2$ possible combinations. Using $1$ rectangle, we can get $7$ solutions. Determining the number of solutions when two rectangles are used appears to be the tricky part. 
 A: With $0$ rectangles there is one way, as you say. However, with $3$ rectangles, there are $3$ ways, not $1$. With $1$ rectangle, there are $7$ ways, as you say.
For $2$ rectangles, they can either both extend in the shorter direction, in $3$ ways, or both in the longer direction, in $4$ ways, or one in each direction, in $4$ ways.
The total is $1+3+7+3+4+4=22$.
[Edit in response to comment:]
Let's say the $2\times3$ rectangle has $2$ rows and $3$ columns. For $2$ rectangles, if they both extend along the columns, they occupy one column each, leaving one column for the squares, and there are $3$ choices for that column. If they both extend along the rows, they have to be in different rows, and each has two possible positions in its row, which makes $2\cdot2=4$ options. If they extend in different directions, one occupies a column, and this can't be the central column, so there are $2$ choices for the column, and in each case there are $2$ choices which row to put the other one in, for another $2\cdot2=4$ options.
