Five-Minute Check (over Lesson 12–2) Mathematical Practices Then/Now New Vocabulary Key Concept: Factorial Example 1: Probability and Permutations of n Objects Key Concept: Permutations Example 2: Probability and nPr Key Concept: Permutations with Repetition Example 3: Probability and Permutations with Repetition Key Concept: Circular Permutations Example 4: Probability and Circular Permutations Key Concept: Combinations Example 5: Probability and nCr Lesson Menu

A spinner is divided into 8 equal sections that are numbered 1 through 8 as shown. Let A be the event that the spinner lands on an odd number. Let B be the event that it lands on a number less than 5. Find A  B. A. {1, 3, 5} B. {1, 2, 3, 4, 7} C. {1, 2, 3, 4, 5, 7} D. {1, 2, 3, 4, 5, 6, 7, 8} 5-Minute Check 1

A spinner is divided into 8 equal sections that are numbered 1 through 8 as shown. Let A be the event that the spinner lands on an odd number. Let B be the event that it lands on a number less than 5. Find P(A  B) A. B. C. D. 5-Minute Check 2

A container has markers in 6 different colors A container has markers in 6 different colors. There are a total of 50 markers in the container. If 8 markers are red, what is the probability of choosing a marker at random and not choosing a red marker? A. B. C. D. 5-Minute Check 3

Kylie surveyed some of her friends to find out if they had gone to the movie theater (event A) or to the football game (event B) over the weekend. The Venn diagram shows the results. What is P(A  B)? A. B. C. D. 5-Minute Check 4

For her birthday, Trina received a new wardrobe consisting of 6 shirts, 4 pairs of pants, 2 skirts, and 3 pairs of shoes. How many new outfits can she make? A. 144 B. 130 C. 94 D. 72 5-Minute Check 5

Mathematical Practices 1 Make sense of problems and persevere in solving them. 4 Model with mathematics. Content Standards S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems. MP

You used the Fundamental Counting Principle. Use permutations with probability. Use combinations with probability. Then/Now

permutation factorial circular permutation combination Vocabulary

Concept

Probability and Permutations of n Objects TALENT SHOW Eli and Mia, along with 30 other people, sign up to audition for a talent show. Contestants are called at random to perform for the judges. What is the probability that Eli will be called to perform first and Mia will be called second? Step 1 Find the number of possible outcomes in the sample space. This is the number of permutations of the order of the 30 contestants, or 30!. Step 2 Find the number of favorable outcomes. This is the number of permutations of the other contestants given that Eli is first and Mia is second, which is (30 – 2)! or 28!. Example 1

Step 3 Calculate the probability. Probability and Permutations of n Objects Step 3 Calculate the probability. number of favorable outcomes number of possible outcomes 1 Expand 30! and divide out common factors. 1 Simplify. Answer: Example 1

Hila, Anisa, and Brant are in a lottery drawing for housing with 40 other students to choose their dorm rooms. If the students are chosen in random order, what is the probability that Hila is chosen first, Anisa second, and Brant third? A. B. C. D. Example 1

Concept

Probability and nPr There are 12 puppies for sale at the local pet shop. Four are brown, four are black, three are spotted, and one is white. What is the probability that all the brown puppies will be sold first? Step 1 Since the order that the puppies are sold is important, this problem relates to permutation. The number of possible outcomes in the sample space is the number of permutations of 12 puppies taken 4 at a time. 1 1 Example 2

Probability and nPr Step 2 The number of favorable outcomes is the number of permutations of the 4 brown puppies in their specific positions. This is 4! or 24 favorable outcomes. Step 3 So the probability of the four brown puppies being sold first is Answer: Example 2

There are 24 people in a hula-hoop contest There are 24 people in a hula-hoop contest. Five of them are part of the Garcia family. If everyone in the contest is equally as good at hula-hooping, what is the probability that the Garcia family finishes in the top five spots? A. B. C. D. Example 2

Concept

Probability and Permutations with Repetition TILES A box of floor tiles contains 5 blue (bl) tiles, 2 gold (gd) tiles, and 2 green (gr) tiles in random order. The desired pattern is bl, gd, bl, gr, bl, gd, bl, gr, and bl. If you selected a permutation of these tiles at random, what is the probability that they would be chosen in the correct sequence? Step 1 There is a total of 9 tiles. Of these tiles, blue occurs 5 times, gold occurs 2 times, and green occurs 2 times. So the number of distinguishable permutations of these tiles is Use a calculator. Example 3

Probability and Permutations with Repetition Step 2 There is only one favorable arrangement— bl, gd, bl, gr, bl, gd, bl, gr, bl. Step 3 The probability that a permutation of these tiles selected will be in the chosen sequence is Answer: Example 3

TILES A box of floor tiles contains 3 red (rd) tiles, 3 purple (pr) tiles, and 2 orange (or) tiles in random order. The desired patter is rd, rd, pr , pr, or, rd, pr, and or. If you selected a permutation of these tiles at random, what is the probability that they would be chosen in the correct sequence? A. B. C. D. Example 3

Concept

Probability and Circular Permutations A. SEATING If 8 students sit at random in the circle of chairs shown, what is the probability that the students sit in the arrangement shown? Explain your reasoning. Since there is no fixed reference point, this is a circular permutation. So there are (8 – 1)! or 7! distinguishable permutations of the way the students can sit. Example 4

Probability and Circular Permutations Answer: The probability of the students sitting in the arrangement shown is Example 4

Probability and Circular Permutations B. CRAYONS You purchase a box of 8 crayons. If the crayons are packaged in random order, what is the probability that the crayon on the far left is red? Explain your reasoning. Since the crayons are packaged in a row, instead of a circle with no fixed reference point, this is a linear permutation. In that case, since there are 8 positions and 1 red crayon, the probability that the crayon on the far left is red is Answer: Example 4

A. TABLE SETTINGS If for a birthday party there are 5 people having cake, and there are 5 different colored plates, what is the probability that if chosen at random the plates will be displayed as seen in the order at the right? A. B. C. D. Example 4

B. CONSTRUCTION A home builder is constructing 6 different models of homes on a major cross street, 5 of which are 2-floored homes, and only 1 home that is 1 floor. If built at random, what is the possibility the 1-floored home will be on the 1st plot of land? A. B. C. D. Example 4

Concept

Probability and nCr A set of alphabet magnets are placed in a bag. If 5 magnets are drawn from the bag at random, what is the probability that they will be the letters a, e, i, o, and u? Step 1 Since the order in which the magnets are chosen does not matter, the number of possible outcomes in the sample space is the number of combinations of 26 letters taken 5 at a time, 26C5. Example 5

Step 3 So, the probability of just getting a, e, i, o, and u is Probability and nCr Step 2 There is only one favorable outcome that all 5 letters are a, e, i, o, and u. The order in which they are chosen is not important. Step 3 So, the probability of just getting a, e, i, o, and u is Answer: Example 5

A set of alphabet magnets are placed in a bag A set of alphabet magnets are placed in a bag. If 4 magnets are drawn from the bag at random, what is the probability that they will be the letters m, a, t, and h? A. ans B. ans C. ans D. ans Example 5