A square grid is separated into 16 small squares ( a diagram of a 4 by 4 square is given but I can't seem to insert diagrams here - each corner is identified as A B C and D and diagonal line from AC )
How many ways can two squares be shaded so that the grid has symmetry about the diagonal AC if :
a) the two squares must not include the diagonal line ?
b) the two squares must include the diagonal line ?
A _____________ B
a) One of the shaded squares must be below the diagonal and the other above. There are 6 possibilities for the first square, and once we've chosen that the other is fixed by the symmetry, so 6 possible answers.
b) If one of the squares is on the diagonal, the other must be as well to obtain symmetry, so the question becomes how many ways can we choose 2 squares out of the 4 on the diagonal. Well, there are 4 ways to choose the first square, times 3 to choose the second (once one square is taken), divided by 2 because we've considered each combination twice, so again 6 possible answers.