Martha places the capital letters A, B, C, and D into the 2x4 table shown. Exactly one letter is placed in each cell. She wishes to make sure that in each row and in each 2x2 square, each of the four letters appears exactly once. How many ways can she do this? Note first that in the first row, A, B, C, and D can be placed completely however Martha likes, and therefore there are four factorial, which is 24 ways to do this. Then, in the second row, let's consider the second square. It can't be the same as the first square of the first row, the second square of the first row, or the third square of the first row, because we know that each letter appears in each 2x2 square once. Therefore, it must be the exact same as the last square of the first row, and then the second square of the second row is exactly determined. This is true of the other positions, and then we know that the bottom row is completely fixed after we determine the first row. Thus, the total number of ways is 24.

Martha

2x4 table

capital letters

permutation

combinatorics