Problem:
Let , and let and be randomly chosen (not necessarily distinct) functions from to . The probability that the range of and the range of are disjoint is , where and are relatively prime positive integers. Find .
Solution:
For each of the one-element subsets there is function on whose range is , so there are functions on with one-element ranges. For each of the two-element subsets , there are functions on whose range is , so there are functions on with twoelement ranges. For each of the three-element subsets , there are functions on whose range is , so there are functions on with three-element ranges. It follows that the number of ordered pairs of functions on whose ranges are disjoint is . The probability that such a pair of functions is chosen at random is . The requested numerator is .
The problems on this page are the property of the MAA's American Mathematics Competitions