Tiling $41$ unit squares with $L$ tetrominoes and $L$ trominoes

Question

In the given figure, there are $41$ unit squares. We want tile to the figure with $L$-tetrominoes and $L$-trominoes. Determine all possible numbers of usable $L$-tetrominoes? Please, prove your answer wiht supporter figures.

Notes:

1. We can rotate and reflect some of $L$-tetrominoes and $L$-trominoes for the tiling operations. Also, number of $L$-tetrominoes and number of $L$-trominoes can be different.

2. Problem is mine and I have its solution. I have sent for sharing. I hope that you like it.

Answer 1

If there are $p$ trominoes and $q$ tetrominoes, then $3p + 4q = 41$, which has $3$ solutions over the positive integers: $(p,q) \in \{(3,8), (7,5), (11, 2)\}$. Each of these is possible, as shown below:

Both in the tilings and in the positive integer equations, the idea is that we can replace three tetrominoes by four trominoes while still covering the same total area.