Math 1Name ______
Notes: Fundamental Counting Principle
In many real-life problems you want to count the number of possibilities. For instance, suppose you own a small deli. You offer 4 types of meat (ham, turkey, roast beef, and pastrami) and 3 types of bread (white, wheat, and rye). How many choices do your customers have for a meat sandwich?
You have learned about tree diagram and that is one way to answer this question; by creating a tree diagram, as shown below. From the list to the right you can see that there are 12 choices.
Another way to count the number of possible sandwiches is to use the
Fundamental Counting Principle. Because you have 4 choices for meat and 3 choices for bread, the total number of choices is
How can the fundamental counting Principle be used to solve the problem above?
Now try these:
1. How many 10-digit phone numbers can be formed with the prefix 706?
2. How many 7-digit phone numbers can be formed if the first digit cannot be 0 or 1?
3. How many 7-digit phone numbers can be formed if the first digit cannot be 0 or 1 and no digit can be repeated?
4. In how many ways can the 4 call letters of a radio station be arranged if the first letter must be W or K and no letters repeat?
5. A company manufactures cheerleading shoes in 3 colors, 2 styles and 8 sizes. How many different cheer shoes can be made?
6. How many different batting orders does a baseball team of nine players have if the pitcher bats last?
7. At Lakeside, Nick is taking six different classes. Assuming that each of these classes is offered each period, how many different schedules might he have?
8. A golf club manufacturer makes irons with 7 different shaft lengths, 3 different grips, 5 different lies, and 2 different club head materials. How many different irons can the company offer?
9. There are 10 questions on a true/false test. If all questions are answered, in how many different ways can the test be completed?
10. A 4-digit code is to be made, using the numbers 0 to 9. How many different codes are there?
11. If you want to draw 2 cards from a standard deck of 52 cards without replacing them, find how many different draws you could make.
121. Police use photographs of various facial features to help witnesses identify suspects. One basic identification kit contains 195 hairlines, 99 eyes and eyebrows, 89 noses, 105 mouths and 74 chins and cheeks.
a. The developer of the identification kit claims that it can produce billions of different faces. Is this claim correct?
b. A witness can clearly remember the hairline and the eyes and eyebrows of a suspect. How many different faces can be produced with this information?
13. The standard configuration for a Georgia license plate is 3 digits followed by 3 letters.
a. How many different license plates are possible if digits and letters can be repeated?
b. How many different license plates are possible if digits and letters cannot be repeated?
14. How many different lunches could you order if there were 4 different sandwiches, 3 different side orders and 4 different drinks?
15. How many different ice cream sundaes could you order if there were 3 different flavors of ice cream, 4 different sauces, and 2 different toppings?
Think of the last two problems that you worked concerning lunch choices (#14)and ice cream sundaes choices (#15).
16. Suppose you wanted lunch and an ice cream sundae. What would you do to find how many choices would you have? (check you answer at the bottom of the page)
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17. Suppose you wanted either lunch or an ice cream sundae. What would you do to find the number of choices you have? (check you answer at the bottom of the page)
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Answers: 16. If you said multiply 48 by 24, you are correct.17. If you said add 48 and 24, you are correct.
Challenge: A 4-digit code is to be made, using the numbers from 0 to 9. But the code cannot be a number greater than 6999, can’t start with zero, and it must be an odd number. How many different codes are there?