91Ó°ÊÓ

In how many ways can three \(x\) 's, three \(y^{\prime}\) s, and three \(z\) 's be arranged so that no consecutive triple of the same letter appears?

If Zachary rolls a fair die five times, what is the probability that the sum of his five rolls is 20 ?

At a 12-week conference in mathematics, Sharon met seven of her friends from college. During the conference she met each friend at lunch 35 times, every pair of them 16 times, every trio eight times, every foursome four times, each set of five twice, and each set of six once, but never all seven at once. If she had lunch every day during the 84 days of the conference, did she ever have lunch alone?

Compute \(\phi(n)\) for \(n\) equal to (a) \(51 ;\) (b) 420 ; (c) 12300 .

Compute \(\phi(n)\) for \(n\) equal to (a) 5186 ; (b) 5187 ; (c) 5188 .

At an upcoming family reunion, five families - each consisting of a husband, wife, and one child - are to be seated around a circular table. In how many ways can these 15 people be arranged around the table so that no family is seated all together? (Here, as in Example 8.9, two seating arrangements are considered the same if one is a rotation of the other.)