For Exercises 69–72, refer to the following. Like mattresses, car tires need to be rotated so as to promote even wear; wear on a tire varies with wheel position. The three main rotation schemes are shown below.
Question 19.101
71. A tire rotation scheme is designed so that no tire will remain where it was. Such a rearrangement (permutation) of objects is called a derangement. Two more tire rotation schemes are shown below. Are there still more schemes that are derangements? One way to record a tire rotation is to label the original tire positions clockwise from the left front: 1 for left front, 2 for right front, 3 for right rear, and 4 for left rear. Record the results of a scheme by writing in turn where each tire goes; for example, scheme D below produces the derangement 4321 because tire 1 goes to position 4, tire 2 goes to position 3, and so forth.
71.
There are four more derangements: , where CW stands for clockwise and CCW for counterclockwise.