I am to color four squares, 2 red and 2 blue, in a 5 x 5 grid such that squares of the same color do not lie in the same row or column. How many ways are there to do this?
I would like to know if my thought process is correct.
Step 1: Out of the 25 squares, we choose one to color it red. There are 25 ways to do this.
Step 2: We now proceed to color another square red. Since the column and the row of the previous square are unavailable for this purpose, we only have 25 - 9 = 16 squares to choose from. There are hence 16 ways to carry out this step.
Step 3: Now, we choose a square to color it blue. There are no restrictions, so we have 23 available squares i.e. 23 ways to carry out this step.
Step 4a: Suppose both red squares are situated in the union of the row and column of the blue square. Then, we have 16 squares to choose from to color the next blue square.
Step 4b: Suppose only one red square is situated in the union of the row and column of the blue square. Then, we only have 15 squares to choose from.
Step 4c: Suppose both red squares are not situated in the union of the row and column of the blue square. Then, we only have 14 squares to choose from.
Now, since steps 4a, 4b and 4c are independent cases, the number of ways to carry out step 4 is 16 + 15 + 14 = 45.
As the entire coloring process is a sequence of steps 1 to 4, the total number of ways to color 4 squares is 25 x 16 x 23 x 45 = 414000. Is this correct?