3. (17 points) In a $3 \times 4$ grid, 4 crosses need to be placed such that each row and each column contains at least one cross. How many ways can this be done?
# Answer: 36

Solution. From the condition, it follows that in some row, two cells are marked (while in the others, only one each). This row can be chosen in 3 ways, and the two crosses in it can be placed in $4 \cdot 3 / 2=6$ ways. The remaining two crosses can be chosen in 2 ways. By the multiplication rule, the total will be $3 \cdot 6 \cdot 2=36$ variants.