Solution below.

Reveal solution by highlighting:

Fibonacci time! We can construct all the 5×2 tilings from the 4×2 and 3×2 tilings. Either we take a 3×2 tiling (and there are 3 of these) and add 2 horizontal pieces, or we take a 4×2 tiling and add a vertical piece. These are all the solutions! So there are 3+5 = 8 5×2 tilings.

F(4) = 3, F(5) = 5. There are F(5) = F(4+1) 4×2 tilings. To find the 12×2 tilings, we want F(13) = 233.