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Notes for math 141 Section 6.4-7.1 Finite Mathematics 6.4: Permutations and Combinations Permutation Definition: A permutation of the set is an arrangement of these objects in a definite order. Example 1: Let A = {a, b, c}, a. Find the numbers of permutations of A b. List all the permutations of A with the aid of a tree diagram. Example 2: Suppose we want to seat 12 people in a row of 12 seats. How many arrangements are possible? Definition: The above product is called a factorial: n! = n(n − 1)(n − 2)⋯3 ⋅ 2 ⋅ 1 Note: 0! = 1 1 Notes for math 141 Section 6.4-7.1 Finite Mathematics Definition: If we have n distinct elements and we want to take rof them in an arrangement, we say that the number of permutations of n things taken r at a time is: n! P (n, r) = n(n − 1)(n − 2)⋯(n − r + 1) = (n − r)! Example 3: How many ways can we select 5 people from a group of 12 and arrange them in 5 chairs? Definition: Permutations of n objects, not all distinct: Given a set of n objects in which n1 are alike of one kind, n2 are alike of another,⋯, nr alike of another so that n1 + n2 + ⋯ + nr = nthen the number of permutations of the n objects taken n at a time is: n! n1 !n2 !⋯nr ! Example 4: Suppose we have 2 red marbles, 3 green marbles, and 1 blue marble. If we want to line the marbles up in a row, how many distinguishable arrangements of the 6 marbles are there? 2 Notes for math 141 Section 6.4-7.1 Finite Mathematics Example 5: How many distinguishable arrangements can we make from the letters in the word Mississippi? Example 6: The twelve directors of a company are to be divided equally into three separate committees to study sales, recent products, and labor relations. In how many ways can this be done? 3 Notes for math 141 Section 6.4-7.1 Finite Mathematics Combinations Definition: A combination is a way of selecting r objects from a set of n objects without any regard to the order. Definition: The number of combinations of n items taken r at a time is: C(n, r) = n! (n − r)!r! Example 7: How many ways are there if we choose 3 people from a group of ten people? Example 8: 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? 4 Notes for math 141 Section 6.4-7.1 Finite Mathematics Example 9: Suppose we have a bag containing 6 purple, 3 red, and 7 green candies. You choose 5 pieces at random. 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? 5 Notes for math 141 Section 6.4-7.1 Finite Mathematics Example 10: 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 10: In how many ways can a committee be formed with a chair, a secretary, a treasurer, and four additional people if they are to all be chosen from a group of ten people? 6 Notes for math 141 Section 6.4-7.1 Finite Mathematics Example 11: An investor has selected a mutual fund to invest his money in. He plans on observing its performance over the next ten years. He will consider the year a success (S) if the mutual fund performs above average and a failure (F) otherwise. a) How many different outcomes are possible? b) How many different outcomes have exactly six successes? c) How many different outcomes have at least three successes? 7 Notes for math 141 Section 6.4-7.1 Finite Mathematics 7.1: Experiments , Sample Spaces and Events Definition: An experiment is an activity with an observable result. Definition: The outcome is the result of an experiment. Definition: The sample space is the set of all possible outcomes of an experiment. Definition: An event is a subset of the sample space. (Note: An event E is said to occur whenever E contains the observed outcome.) Definition: E and F are mutually exclusive if E ∩ F = ∅. Note: All of the set operations (union, intersection, complement) work the same with events. 8 Notes for math 141 Section 6.4-7.1 Finite Mathematics Example 1: Lets consider the experiment of rolling a fair six-sided die and observing the number that lands uppermost. a) Determine the sample space. b) Find the event E where E = {x∣x is an even number}. c) Find the event F where F = {x∣x is a number greater than 2} d) Find the event (E ∩ F ) e) Find the event (E ∪ F ) f) Are the events E and F mutually exclusive? g) Are the events E and F complementary? 9 Notes for math 141 Section 6.4-7.1 Finite Mathematics Example 2: Lets consider the experiment of flipping a fair coin two times and observing the resulting sequence of heads and tails. a) Determine the sample space. b) Find the event E where E = {x∣x has one or more heads} c) Find the event F where E = {x∣x has more than 2 heads} d) List all events of this experiment. (Note: we can use tree diagrams to determine all the possible outcomes. ) 10 Notes for math 141 Section 6.4-7.1 Finite Mathematics Example 3: An experiment consists of casting a pair of fair six sided dice and observing the number that falls uppermost on each die. a) Find the sample space for this experiment. b) Determine the event that the sum of the numbers falling uppermost is less than or equal to 6. c) Determine the event that the number falling uppermost on one die is a 4 and the number falling uppermost on the other die is greater than 4. 11