91Ó°ÊÓ

In how many ways can Troy select nine marbles from a bag of twelve (identical except for color), where three are red, three blue, three white, and three green?

Ten women attend a business luncheon. Each woman checks her coat and attaché case. Upon leaving, each woman is given a coat and case at random. (a) In how many ways can the coats and cases be distributed so that no woman gets either of her possessions? (b) In how many ways can they be distributed so that no woman gets back both of her posses. sions?

Ms. Pezzulo teaches geometry and then biology to a class of 12 advanced students in a classroom that has only 12 desks. In how many ways can she assign the students to these desks so that (a) no student is seated at the same desk for both classes? (b) there are exactly six students each of whom occupies the same desk for both classes?

a) In how many ways can the integers \(1,2,3, \ldots, n\) be arranged in a line so that none of the patterns \(12,23,34, \ldots,(n-1) n\) occurs? b) Show that the result in part (a) equals \(d_{n-1}+d_{n}\). \(\left(d_{n}=\right.\) the number of derangements of \(1,2,3, \ldots, n .)\)

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

For which positive integers \(n\) is \(\phi(n)\) a power of \(2 ?\)