Objective: In this lesson, you will use permutations and combinations to compute probabilities of compound events and to solve problems. Read this knowledge article and answer the following: Tree Diagrams and the Fundamental Counting Principle Example 1 If a woman has two blouses {B1, B2} and three skirts {s1, s2, s3}, how many different outfits consisting of a blouse and a skirt can she wear? The fundamental counting principle states that if one task can be done in m ways, and a second task can be done in n ways, then Example 2 A truck license plate consists of a letter followed by four digits. How many such license plates are possible? Determining Probability with the Fundamental Counting Principle Example 3 A truck license plate consists of a letter followed by four digits. If you select one license plate at random, what is the probability it contains the letter Z? Page 1
Permutations Find the number of sequences that can be formed with the letters {A. B, C} if no letter is repeated. Draw a tree diagram to determine the number of sequences. The tree diagram gives us Use the fundamental counting principle to determine the number of sequences. We have three choices for the, two choices for the, and one choice for the. The fundamental counting principle gives us. We can write this equation using factorial notation as Factorial notation is defined as n! = 1! = 1 and 0! = 1 Where order is important and no element is repeated are permutations. A permutation of a set of elements is an ordered arrangement where each element is used once. Example 4 Given five letters {A, B, C, D, E}, find the number of four-letter sequences possible. Permutations of n objects taken r at a time is written as. Example 4 can also be answered as: The number of four-letter sequences is = 5P4 = 120. Page 2
Example 6 You have 4 math books and 5 history books to put on a shelf that has 5 slots. In how many ways can the books be shelved if the first three slots are filled with math books and the next two slots are filled with history books? First we determine the number of ways the math books can be arranged. This is a permutation of 4 objects taken 3 at a time, or 4P3: 4P3 = Next we determine the number of ways the history books can be arranged. This is a permutation of 5 objects taken 2 at a time, or 5P2: 5P2 = Since there are 24 ways to arrange the math books and 20 ways to arrange the history books, the total number of ways to arrange the math and history books is Page 3
Combinations Suppose that we have a set of three letters {A, B, C}, and we are asked to make two-letter sequences. We have six permutations: AB BA BC CB AC CA Now suppose we have a group of three people {A, B, C}, Al, Bob, and Chris, respectively, and we are asked to form committees of two people each. This time we have only three committees: AB BC AC When forming committees, the order is not important because the committee that has Al and Bob is no different from the committee that has Bob and Al. As a result, we have only three committees, not six. Forming sequences is an example of permutations. Forming committees is an example of combinations. Permutations are those arrangements where Combinations are those arrangements where ncr represents the number of combinations of n objects taken r at a time. Page 4
Combinations Involving Several Sets Example 8 How many five-person committees consisting of 2 men and 3 women can be chosen from a group of 4 men and 4 women? First we consider the number of combinations of 4 men chosen 2 at a time: Next we consider the number of combinations of 4 women chosen 3 at a time: 6 4 = Probabilities Involving Combinations Example 9 A high school club consists of 4 freshmen, 5 sophomores, 5 juniors, and 6 seniors. What is the probability that a committee of 4 people chosen at random includes one student from each class? The total number of committees possible is 20C4 since there are 20 students in all and we want to select 4: 20C4 = 4,845 The number of combinations of 4 freshman taken 1 at a time is The number of combinations of 5 sophomores taken 1 at a time is The number of combinations of 5 juniors taken 1 at a time is The number of combinations of 6 juniors taken 1 at a time is 4 5 5 6 = Example 10 A high school club consists of 4 freshmen, 5 sophomores, 5 juniors, and 6 seniors. What is the probability that a committee of 4 people chosen at random includes at least one senior? P (at least one senior) = 0.451 + 0.282 + 0.058 + 0.003 = Page 5
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