I'm stuck with this question
We have eight $1\times 2$ tiles that each one of them has one $1\times 1$ blue square and one $1\times 1$ red square. We want to cover a $4\times 4$ area with these tiles in a way that every row and every column of this area had exactly $2$ blue and $2$ red $1\times 1$ squares. In how many ways we can do this?
First of all, I tried to find out all possible ways which I can cover a $4\times 4$ square by $1\times 2$ dominoes. If I know all such possibilities, then for each such a covering, by obtaining the number of ways that some one can tiles this dominoes in blue-red squares in the manner that described above, we can get the answer. In the article How Many Ways Can We Tile a Rectangular Chessboard With Dominoes? the writer claims that the number of ways which we can tile a $4\times 4$ rectangle is $36$. But he did not described all this $36$ ways.
The above method is very prolix and also, I need to obtain all the possibilities of covering $4\times 4$ square by $1\times 2$ tiles and for each of such covering, all the ways which can satisfied by the conditions of question. Dose any one know a simpler method?
I should point out that the above question is designed for high school students and so I think it must have a simple solution.