A cycle lock has three wheels, each showing one of the numbers 1 to 6, and every setting is equally likely. What is the probability that EXACTLY two of the three wheels show the same number?

There are $6\times 6\times 6=216$ settings. For exactly two alike, choose the repeated number in 6 ways, choose a different number for the odd wheel in 5 ways, and choose WHICH of the three wheels is the odd one in 3 ways: $6\times 5\times 3=90$ settings, a probability of 90/216 = 5/12. The value 5/36 = 30/216 comes from $6\times 5$ with the three positions forgotten, the commonest slip here. The value 5/9 = 120/216 counts settings with all three wheels different, from $6\times 5\times 4$, and 1/36 = 6/216 counts all three the same. Those three cases exhaust the possibilities, and as a check 5/12 + 5/9 + 1/36 = 15/36 + 20/36 + 1/36 = 1.