Combinatorics 5Try problem with * later.

P1 Mr. and Mrs. Zeta want to name their baby Zeta so that itsmonogram (first, middle, and last initials) will be in alphabeticalorder with no letters repeated. How many such monograms arepossible?

P2 The student lockers at Olympic high are numered consecutivelybeginning with locker number 1. The plastic digits used to numberthe lockers cost two cents a piece. Thus, it costs two cents to labellocker number 9 and four cents to label locker number 10. If it costs$137.94 to label all the lockers, how many lockers are there at theschool?

P3 Let n be an odd integer greater than 1. prove that the sequence(n1

),(n2

), · · · ,

(n

n−12

)contains an odd number of odd numbers.

P4 How many positive integers not exceeding 2001 are multiples of3 or 4 but not 5?

P5 Let x = ·123456789101112 · · · 998999, where the digits areobtained by writing the integers 1 through 999 in order. Find the1983rd digit to the right of the decimal point.

P6* Twenty five boys and twenty five girls sit around a table. Provethat it is always possible to find a person both of whose neighborsare girls.

P7 At the end of a professional bowling tournament, the top 5bowlers have a play-off. First #5 bowls #4. The loser receives 5th

prize and the winner bowls #3 in another game. The loser of thisgame receives 4th prize and the winner bowls #2. The loser of thisgame receives 3rd prize and the winner bowls #1. The winner ofthis game gets 1st prize and the loser gets 2nd prize. In how manyorders can bowlers #1 through #5 receive the prizes?

P8 A spider has one sock and one shoe for each of its eight legs.In how many different orders can the spider put on its socks andshoes, assuming that, on each leg, the sock must be put on beforethe shoe?

P9* A drawer in a darkened room contains 100 red socks, 80 greensocks, 60 blue socks and 40 black socks. A youngster selects socksone at a time from the drawer but is unable to see the color of thesocks drawn. What is the smallest number of socks that must be

selected to guarantee that the selection contains at least 10 pairs?(A pair of socks is two socks of the same color. No sock may becounted in more than one pair.)

P10 Given a rational number, write it as a fraction in lowest termsand calculate the product of the resulting numerator and denomi-nator. For how many rational numbers between 0 and 1 will 20! bethe resulting product?

P11 Determine the number of ways to choose five numbers fromthe first eighteen positive integers such that any two chosen num-bers differ by at least 2.

P12 In a room containing N people, N > 3, at least one personhas not shaken hands with everyone else in the room. What is themaximum number of people in the room that could have shakenhands with everyone else?

P13* Find the number of ordered quadruples (x1, x2, x3, x4) of pos-itive odd integers that satisfy x1 + x2 + x3 + x4 = 98.

P14* Finitely many cards are placed in two stacks, with more cardsin the left stack than in the right. each card has one or more dis-tinct names written on it, although different cards may share somenames. For each name, we define a shuffle by moving every cardthat has that name written on it to the opposite stack. Prove thatit is always possible to end up with more cards in the right stackby picking several distinct names, and doing in turn the shuffle cor-responding to each name.

P15 For how many pairs of consecutive integers in the set{1000, 1001, 1002, · · · , 2000} is no carrying required when the twointegers are added?

P16 Nine chairs in a row are to be occupied by six students andProfessors Alpha, Beta, and Gamma. These three professors arrivebefore the six students and decide to choose their chairs so thateach professor will be between two students. In how many wayscan Professors Alpha, Beta, and Gamma choose their chairs?

P17* Prove that among any 16 distinct positive integers not ex-ceeding 100 there are four different ones, a, b, c, d, such that a+ b =c + d.