Find the length. Round to the nearest hundredth. QR Warm Up Need a calculator 12.9(sin 63 ) = QR 11.49 cm QR
Check Homework
Objectives Solve problems involving permutations.
For a main dish, you can choose steak or chicken; your side dish can be rice or potatoes; and your drink can be tea or water. Make a tree diagram to show the number of possible meals if you have just one of each.
A sandwich can be made with 3 different types of bread, 5 different meats, and 2 types of cheese. How many types of sandwiches can be made if each sandwich consists of one bread, one meat, and one cheese. Method 1 Use a tree diagram. Bread 1 2 Meat Cheese 1 2 3 4 5 1 2 1 2 1 2 1 2 1 2 1 2 3 4 5 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 There are 30 possible types of sandwiches. 3 1 2 3 4 5
Sometimes there are too many possible outcomes to make a tree diagram or a list. The Fundamental Counting Principle is one method of finding the number of possible outcomes.
A sandwich can be made with 3 different types of bread, 5 different meats, and 2 types of cheese. How many types of sandwiches can be made if each sandwich consists of one bread, one meat, and one cheese. Method 2 Use the Fundamental Counting Principle. 3 5 2 There are 3 choices for the first item, 5 choices for the second item, and 30 2 choices for the third item. There are 30 possible types of sandwiches.
A florist is arranging centerpieces that include 1 flower, 1 plant, and 1 vase. The florist has 2 kinds of vases, 2 kinds of plants, and 3 kinds of flowers to choose from. How many different centerpieces are possible? Method 1: Make a tree diagram Method 2: Use the Fundamental Counting Principle 2 X 2 X 3 = 12
A voicemail system password is 1 letter followed by a 3-digit number less than 600. How many different voicemail passwords are possible? Use the Fundamental Counting Principle. 26 600 15,600 There are 26 choices for letters and 600 different numbers (000-599). There are 15,600 possible combinations of letters and numbers.
Use the Fundamental Counting Principle to find the total number of outcomes in each situation. 1. choosing north, south, east, or west and one of the 50 states 2. picking a day of the week and a month of the year 3. choosing vanilla, strawberry, chocolate, or mint chip ice cream with fudge, butterscotch, strawberry, or whipped topping, in a cone or a cup 4. Margarita wants to wear a different outfit to school each day using her new clothes. Margarita bought 5 pairs of pants, 9 shirts, and 4 pairs of shoes. How many days of school does Margarita expect to have? 5. Ryan has a business screen printing T-shirts. Ryan offers 12 color options, 3 T-shirt styles, and printing in 1, 2, 3, 4, or 5 colors. How many different styles of T-shirts does Ryan s business offer?
ANSWERS 1. 200 2. 84 3. 32 4. 180 5. 180
Permutation a selection of objects from a group in which order is important
Factorials The factorial of a number is the product of the number and all the natural numbers less than the number. The factorial of 4 is written 4! and is read four factorial. 4! = 4 3 2 1 = 24. The factorial of 5 is written 5! and is read five factorial. 5! = 5 4 3 2 1 = 120.
Interesting The factorial of 0 is defined to be 1. 0! = 1
There are 7 members in a club. What is the number of permutations of the 7 members of the club? (Think about lining up all of the members of the club- order matters here).
There are 7 members in the club. The club is holding elections for a president, a vice president, and a treasurer. How many different ways can these positions be filled?
What is the number of permutations of the members of the club who were not elected as officers?
Divide the number of permutations of all the members by the number of permutations of the unelected members. Compare this number to the number of permutations of elected officers.
Explain the effect of dividing the total number of permutations by the number of permutations of items not selected.
Suppose you want to make a five-letter password from the letters A, B, C, D, and E without repeating a letter. You have 5 choices for the first letter, but only 4 choices for the second letter. You have one fewer choice for each subsequent letter of the password. Order of the letters matter for the password! or we can say 5! = 120
Suppose you want to make a three-letter password from the 5 letters A, B, C, D, and E without repeating a letter. Again, you have one fewer choice for each letter of the password. The number of permutations is:
A group of 8 swimmers are swimming in a race. Prizes are given for first, second, and third place. How many different outcomes can there be? The order in which the swimmers finish matters so use the formula for permutations. n = 8 and r = 3. A number divided by itself is 1, so you can divide out common factors in the numerator and denominator. There can be 336 different outcomes for the race.
How many different ways can 9 people line up for a picture? The order in which the people line up matters so use the formula for permutations. = 362,880 n = 9 and r = 9. A number divided by itself is 1, so you can divide out common factors in the numerator and denominator. There are 362,880 ways the 9 people can line up for the picture.
1. How many permutations are possible of the letters in the word secret? 2. Julie, Dan, Janet, Kevin, and Michael all enter a contest. Two names are pulled out of a hat one at a time. In how many ways can the contest winners be selected? 3. A child has the magnetic letters, V, O, L, E. In how many ways can the letters be arranged? 4. Carlos, Sierra, and Nicole go to the movies and sit in a row of three seats. How many seating arrangements are possible? 5. In how many ways can a football coach arrange the first five players in a lineup of eleven players?
ANSWERS 1. 720 2. 20 3. 24 4. 6 5. 55, 440
Homework NONE