Apply the Counting Principle and Permutations Section 10

Apply the Counting Principle and Permutations Section 10. 1

Objectives: �Use the fundamental counting principle �Use permutations with and without repetition

Key Vocabulary: �Permutation �Factorial

Example 1: A sporting goods store offers 3 types of snowboards (all-mountain, freestyle, and carving) and 2 types of boots (soft and hybrid. How many choices does the store offer for snowboarding equipment?

Fundamental Counting Principle

Example 2: You are framing a picture. The frames are available in 12 different styles. Each style is available in 55 different colors. You also want blue mat board, which is available in 11 different shades of blue. How many different ways can you frame the picture?

Example 3: The standard configuration for a Texas license plate is 1 letter followed by 2 digits followed by 3 letters. a) How many different license plates are possible if letters and digits can be repeated? b) How many different license plates are possible if letters and digits cannot be replaced?

#1: Complete. The store in Example 1 also offers 3 different types of bicycles (mountain, racing, and BMX) and 3 different wheel sizes (20 in. , 22 in. , and 24 in. ). How many bicycle choices does the store offer?

#2: Complete. In Example 3, how do the answers change for the standard configuration of a New York license place, which is 3 letters followed by 4 numbers?

Permutations A permutation is an arrangement of objects in which order is important. For instance, the 6 possible permutation of the letters A, B, and C are shown: ABC ACB CBA BAC BCA CAB

Factorial For any positive integer n, the product of the integers from 1 to n is called n factorial and it written as n! The value of 0! Is defined to be 1.

Example 4: Ten teams are competing in the final round of the Olympic four-person bobsledding competition. a) In how many different ways can the bobsledding teams finish the competition? (Assume there are no ties. ) b) In how many different ways can 3 of the bobsledding teams finish first, second, and third to win the gold, silver, and bronze medals?

#3: Complete. In Example 4, how would the answers change if there were 12 bobsledding teams competing in the final round of the competition?

Permutations

Example 5: You are burning a demo CD for your band. Your band has 12 songs stored on your computer. However, you want to put only 4 songs on the demo CD. In how many orders can you burn 4 of the 12 songs onto the CD?

#4: Find the number of permutations.

#5: Find the number of permutations.

#6: Find the number of permutations.

#7: Find the number of permutations.

Permutations with Repetition

Example 6: Find the number of distinguishable permutation of the letters in a) MIAMI b) TALLAHASSEE

#8: Find the number of distinguishable permutations of the letters in the word. MALL

#9: Find the number of distinguishable permutations of the letters in the word. KAYAK

#10: Find the number of distinguishable permutations of the letters in the word. CINCINNATI

Homework Assignment Page 686 #4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 44, 46, 48, 50, 52, 54, 62, 66