A gardener has a circular garden, which he divides into four quadrants. He has four different kinds of flowers, and he wants to assign a flower to each quadrant, so that adjacent quadrants are not planted with the same kind of flower. (Also, the gardener can use the same flower more than once.) How many different gardens are possible? Answer Not 24
The 1 must go in the upper-left box, and the 6 must go in the lower-right box. There are still two boxes left in the first row. There are four numbers left to choose from, namely 2, 3, 4, and 5. There are six ways to choose two numbers from these four numbers, namely 2 and 3, 2 and 4, 2 and 5, 3 and 4, 3 and 5, and 4 and 5. Once these numbers are chosen, the smaller number must go on the left, and the larger number must go on the right. Then the remaining three numbers must fill the second row, in increasing order. Of all six arrangements, only the arrangement in the lower-right corner does not work (a 4 is above a 3), so there are 5 possible arrangements.