Problem:
Suppose that boys and girls line up in a row. Let be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row we have . The average value of (if all possible orders of these people are considered) is closest to
Answer Choices:
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Solution:
Suppose that John and Carol are two of the people. For , let and be the numbers of orderings (out of all ) in which the and persons are John and Carol, or Carol and John, respectively. Then is the number of orderings of the remaining persons.
For , let be the number of times a boy-girl or girl-boy pair occupies positions and . Since there are boys and girls, . Thus the average value of is
In general, suppose there are boys and girls. For let be the probability that there is a boy-girl pair in positions in the line. Since there is either or pair in is also the expected number of pairs in these positions. By symmetry, all 's are the same (or note that the argument below is independent of ). Thus the answer is .
We may consider the boys indistinguishable and likewise the girls. (Why?) Then an order is just a sequence of Bs and Gs. To have a pair at we must have BG or GB in those positions, and the remaining positions must have boys and girls. Thus there are sequences with a pair at . Since there are sequences,
In our case, the answer is .