cKendra Kilmer September 19, 2008Section 2.1 - The Multiplication Principle and PermutationsExample 1: A yogurt shop has 4 flavors (chocolate, vanilla, strawberry, and blueberry) and three sizes (small,medium, and large). How many different yogurts can be ordered?Multiplication Principle: Suppose a task T1can be performed in N1ways, a task T2can be performed in N2ways,. . . , and a task Tncan be performed in Nnways. Then, the number of ways of performing the tasks T1,T2,. . . , Tnin succession is given by the product:N1N2· · · NnExample 2: A coin is tossed 3 times, and the sequence of heads and tails is recorded. Determine the number ofoutcomes.Example 3: An auto manufacturer has 3 different subcompact cars in the line. Customers selecting one of thesecars have a choice of 3 engine sizes, 4 body styles, and 3 color schemes. How many different selections can acustomer make?1cKendra Kilmer September 19, 2008Example 4: How many three-letter words that have exactly one vowel can be made using the first seven lettersof the alphabet where using a letter twice is permitted but having two consonants next to each other is not?Example 5: How many five-digit numbers can be formed ifa) Zero is not the first digit?b) Zero cannot be the first digit and no digit can be repeated?c) Zero cannot be the first digit, no digit can be repeated, and each number formed must be even?2cKendra Kilmer September 19, 2008Example 6: Jack and Jill and 5 of their friends go to the movies. They all sit next to each other in the same row.How many ways can this be done ifa) there are no restrictions?b) Jill must sit in the middle?c) Jill sits on one end of the row and Jack sits on the other end of the row?d) Jack, Jill, or John sit in the middle seat?e) Jack, Jill, and John sit in the middle seats?f) Jack and Jill must sit next to each other?g) Jill must not sit next to Jack?3cKendra Kilmer September 19, 2008Example 7: Suppose we want to seat 12 people in a row of 12 seats. How many arrangements are possible?The above product is called a factorial:n! = n(n − 1)(n − 2)· · · 3 · 2 · 1Note: 0! = 1Example 8: How many ways can we select 5 people from a group of 12 and arrange them in 5 chairs?Definition: If we have n distinct elements and we want to take r of them in an arrangement, we say that thenumber of arrangements of n things taken r at a time is:n!(n − r)!Example 9: How many ways can we select 25 people from a group of 35 and arrange them in 25 chairs?4cKendra Kilmer September 19, 2008Arrangement of n objects, not all distinct: Given a set of n objects in which n1are alike of one kind, n2arealike of another,. . ., nralike of another so that n1+ n2+ · · · + nr= n then the number of arrangements of the nobjects taken n at a time is:n!n1!n2!· · · nr!Example 10: Suppose we have 2 red marbles, 3 green marbles, and 1 blue marble. If we want to line the marblesup in a row, how many distinguishable arrangements of the 6 marbles are there?Example 11: How many distinguishable arrangements can we make from the letters in the word Mississippi?Example 12: The twelve directors of a company are to be divided equally into three separate committees to studysales, recent products, and labor relations. In how many ways can this be done?5cKendra Kilmer September 19, 2008Section 2.2 - CombinationsDefinition: The number of combinations of n items taken r at a time is:C(n, r) =n!(n − r)!r!Example 1:Example 2: Suppose a high school choir made of 11 students decides to send 2 members to a duet competition.a) How many pairs are possible?b) If it is decided that one particular member is to go, how many different pairs are possible?c) If there are 8 girls and 3 boys in the choir, how many pairs will include at least one boy?6cKendra Kilmer September 19, 2008Example 3: Suppose we have a bag containing 6 purple, 3 red, and 7 green candies. You choose 5 pieces atrandom.a) How many samples of 5 candies can be chosen?b) How many samples are there in which all the candies are green?c) How many samples are there in which they are all red?d) How many samples are there in which there are 2 purple and 1 red?e) How many samples are there in which there are no purple candies?f) How many samples contain at least 1 purple?g) How many samples contain exactly 2 purple or exactly 2 green candies?7cKendra Kilmer September 19, 2008Example 4: Suppose we are playing the lottery in which we must choose 6 from 50 numbers.a) How many different lottery picks could we choose if the order we choose our numbers in does not matter?b) How many ways are there to choose no winning numbers?c) How many ways are there to choose at least 3 winning numbers?Example 5: In how many ways can a committee be formed with a chair, a secretary, a treasurer, and fouradditional people if they are to all be chosen from a group of ten people?8cKendra Kilmer September 19, 2008Example 6: An investor has selected a mutual fund to invest his money in. He plans on observing its performanceover the next ten years. He will consider the year a success (S) if the mutal fund performs above average and afailure (F) otherwise.a) How many different outcomes are possible?b) How many different outcomes have exactly six successes?c) How many different outocmes have at least three successes?Example 7: Five cards are randomly selected from a standard deck of 52 cards to form a poker hand. Determinethe number of ways a person can be dealt a full house (that is a three of a kind and a two of a kind)9cKendra Kilmer September 19, 2008Section 2.3 - Probability Applications of Counting PrinciplesComputing the probability of an event in a uniform sample space:Let S be a uniform sample space and let E be any event. Then,P(E) =n(E)n(S)where n(E) is the number of outcomes in E and n(S) is the number of outcomes in S.Example 1: Four marbles are selected at random without replacement from a bowl containing five white andeight green marbles. Find the probability that at least two of the marbles are white.Example 2: An unbiased coin is tossed six times. What is the probability that the coin will land headsa) Exactly three times?b) At most three times?c) On the first and the last toss?10cKendra Kilmer September 19, 2008Example 3: a) An exam consists of ten true-or-false questions. If a student randomly guesses on each question,what is the probability that he or she will answer exactly six questions correctly?b) An exam consists of ten mutliple choice questions each