6. Lil writes one of the letters $\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}$ in each cell of a $2 \times 4$ table. She does this in such a way that, in each row and in each $2 \times 2$ square, all four letters appear. In how many ways can she do this?

A 12
B 24
C 48
D 96
E 198
Solution
B

The letters P, Q, R and S can be arranged freely in the four squares of the first row. There are four options for the first square, then three options for the second square and two for the third square. That is $4 \times 3 \times 2 \times 1$, which is equal to 24 options.
Having filled the first row, consider the second square of the second row. This can't be the same letter as in the first, second or third position of the first row. So it must be the same letter as in the fourth position. There is now only one choice for the letters in the first and third positions of the second row and, once these are chosen, only one choice for the fourth position in that row. Therefore the second row is uniquely defined by the first row and there are 24 options.