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EXAMPLE 1

Snowboarding. 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?. EXAMPLE 1. Use a tree diagram. SOLUTION.

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EXAMPLE 1

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  1. Snowboarding 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? EXAMPLE 1 Use a tree diagram SOLUTION Draw a tree diagram and count the number of branches.

  2. ANSWER The tree has 6 branches. So, there are 6 possible choices. EXAMPLE 1 Use a tree diagram

  3. Photography 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 2 Use the fundamental counting principle

  4. ANSWER The number of different ways you can frame the picture is 7260. = 7260 = 125511 Number of ways EXAMPLE 2 Use the fundamental counting principle SOLUTION You can use the fundamental counting principle to find the total number of ways to frame the picture. Multiply the number of frame styles (12), the number of frame colors (55), and the number of mat boards (11).

  5. License Plates The standard configuration for a Texas license plate is 1 letter followed by 2 digits followed by 3 letters. How many different license plates are possible if letters and digits can be repeated? How many different license plates are possible if letters and digits cannot be repeated? EXAMPLE 3 Use the counting principle with repetition

  6. There are 26choices for each letter and 10choices for each digit. You can use the fundamental counting principle to find the number of different plates. = 26101026262626 Number of plates ANSWER With repetition, the number of different license plates is 45,697,600. EXAMPLE 3 Use the counting principle with repetition SOLUTION = 45,697,600

  7. If you cannot repeat letters there are still 26choices for the first letter, but then only 25remaining choices for the second letter, 24choices for the third letter, and 23choices for the fourth letter. Similarly, there are 10choices for the first digit and 9choices for the second digit. You can use the fundamental counting principle to find the number of different plates. = 26109252423 Number of plates With repetition, the number of different license plates is 32,292,000. ANSWER EXAMPLE 3 Use the counting principle with repetition = 32,292,000

  8. SPORTING GOODS : The store in Example 1 also offers 3 different types of bicycles (mountain, racing, and BMX) and 3 different wheel sizes (20in.,22in., and 24in.). How many bicycle choices does the store offer? ANSWER The store offers 9 choices of bicycles for Examples 1, 2 and 3 GUIDED PRACTICE SOLUTION choices = types · wheel sizes choices =3 · 3 choices =9

  9. What If?In Example 3, how do the answers change for the standard configuration of a New York license plate, which is 3 letters followed by 4 numbers? for Examples 1, 2 and 3 GUIDED PRACTICE

  10. There are 26choices for each letter and 10choices for each digit. You can use the fundamental counting principle to find the number of different plates. = 26262610101010 Number of plates ANSWER The number of plates will increase to 175,760,000. EXAMPLE 3 Use the counting principle with repetition SOLUTION = 175,760,000

  11. If you cannot repeat letters there are still 26choices for the first letter, but then only 25remaining choices for the second letter, 24choices for the third letter. Similarly, there are 10choices for the first digit and 9choices for the second digit and so on. You can use the fundamental counting principle to find the number of different plates. = 26252410987 Number of plates The number of plates would increase to 78,624,000. ANSWER EXAMPLE 3 Use the counting principle with repetition = 78,624,000

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