Probability theory and statistics Academic year 2010/11, II. semester 1 Combinatorics 1.1 You are eating at Emile's restaurant and the waiter informs you that you have (a) two choices for appetizers: soup or juice; (b) three for the main course: a meat, sh, or vegetable dish; and (c) two for dessert: ice cream or cake. How many possible choices do you have for your complete meal? 1.2 Find the number of possible arrangements of 8 castles on the chess board in a way that they do not hit each other? What is the result if we can distinguish between the castles? 1.3 How many real four digit numbers (they can not start with zero) can be formed from digits 0, 1, 2, 3, 4, 5, 6? 1.4 We have 12 books on the shelf. How many ways can the books be arranged on the shelf if 3 particular books must to be next to each other a) if the order of the three books does not count? b) if the order of the three books does count? 1.5 In how many ways can one arrange 4 math books, 3 chemistry books, 2 physics books, and 1 biology book on a bookshelf so that all the math books are together, all the chemistry books are together, and all the physics books are together? 1.6 In how many ways can 7 people be arranged around a round table? 1.7 In how many ways can 5 men and 5 women be arranged around a round table if neither two men, nor two women can sit next to each other? 1.8 In how many ways can one ll a toto coupon (14 matches, three possible results: 1, 2 or X)? 1.9 Three postmen has to deliver six letters. Find the number of possible distribution of the letters. 1.10 Find the number of possible choices of four cards of four different colours from a deck of ordinary cards (4 colours, 13 cards per colour). What is the result if we require that the four cards should be or different gures? 1.11 Find the number of possible llings of a lottery coupon (5 numbers from 90). 1.12 Find the number of possible paths from the origin to the point (5, 3) if we can walk only on points with integer coordinates and we can step only upwards and right. 1 1.13 Starting from origin at each step we toss a coin and in case of a head we make a step to left, otherwise to right. In how many ways can we return to origin in 10 steps? 1.14 Prove the binomial theorem i.e. for all 11,!) E C and n E N we have (a. + b)\" = i (2') a"b""\". (:1:)=(.:1)+G:)- (:)+(':)+-~+(:>=2"- 1.17 Find the number of possible arrangements of n zeros of k ones (k g n+ 1), if two ones can not be next to each other. 1.15 Prove that 1.16 Prove that 1.18 In Circus Maximus the tamer has to lead 5 lions and 4 tigers to the ring, but tigers can not follow each other because they ght. In how many ways can he do if the animals can be distinguished? 1.19 Around the round table of King Arthur 12 knights are sitting. Each of them hates his two neighbours. In how many ways can we choose ve knights without having enemies among them? 1.20 A deck of ordinary cards is shufed and 10 cards are dealt. a In how many cases will we have aces among the 10 cards? b Exactly one ace? ) l c) At most one ace? d) Exactly two aces? e) At least two aces? 1.21 In how many ways we can chose four dancing pairs from 12 girls and 15 boys? 1.22 Find the number of real six digit numbers having three odd and three even digits. 1.23 In how many ways we can distribute 14 persons into four boats with ve, four, three and two seats, respectively? 1.24 In the canteen we can by four types of snacks. In how many ways can we buy 12 of them? 1.25 In how many ways we can distribute 7 apples and 9 peaches among 4 kids? 2 1.26 Peter is putting 10 identical balls in five different buckets. In how many ways can this be done if no bucket is allowed to be empty? 1.27 In how many ways we can choose five cards from a deck with having a club and an ace among them? 1.28 In how many ways we can choose four persons from five boys and five girls having at least two girls among them