Given the following matrix:
$$\begin{bmatrix} a & b & c & d\\ e & f & g & h\\ i & j & k & l \end{bmatrix}$$
how many sets of 3 letter sets using all the letters once can be made if:
- Letters in the same row can't be in the same 3 set.
- As in 1 plus not all 3 letters can be from the same col ( but 2 can ).
For example: $$\{ \{a , g , k \} , \{ b, f , j \} , \{ c, e, l \} , \{ d , h, i \} \}$$ meets the first condition but not the second.
I started out on the first part by choosing pairs for each letter in the first row disregarding order: $$\frac{\binom{8}{2}\binom{6}{2}\binom{4}{2}}{4!}$$
I know I need to use inclusion exclusion for the second half, but I'm not sure how to proceed.