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Counting Principles

Learn how to use the Counting Principles of Multiplication and Addition to solve mathematical problems involving choices and possibilities. Examples and explanations provided to help you grasp the concepts.

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Counting Principles

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  1. Counting Principles Mulitplication Counting Principle & Addition Counting Principle

  2. Multiplication Counting Principle • If one event can occur in m ways, and another event can occur n ways Then the number of ways both can occur together is m x n.

  3. Multiplication Counting Principle • Example 1: • At a store, skateboards are available in 8 different deck designs. Each deck design is available with 4 different wheel assemblies. • How many skateboard choices does the store offer?

  4. Ex1: M.C. Princple • Use the multiplication counting principle • #of diff. designs x # of diff. wheels • 8 x 4 • = 32 skateboard choices

  5. You Try: M.C. Principle • Ex2: Your class is having an election. There are 4 candidates for president, 6 for vice president, 3 for secretary, 7 for treasurer. • How many ways can a president, vice president, secretary, and treasurer be chosen?

  6. EX2: M.C. Principle • Answer: • 4 president x 6 vice pres. x 3 sec. x 7 treasurer • 4 x 6 x 3 x 7 • = 504 different ways

  7. Ex3: Use the multiplication counting principle to find the number of choices that are available • a. Choose sneakers, shoes, or sandals in white, black, or gray • Answer: 3 x 3 = 9 different ways • b. Choose small, medium, large, or extra large pants in dark blue, light blue, or black • Answer: 4 x 3 = 12 different ways

  8. Addition Counting Principle • If the possibilities being counted can be divided into groups with no possibilities in common, then the number of possibilities is the sum of the numbers of the possibilities in each group

  9. Addition Counting Principle • Example 1: I.D. Cards • Suppose that each student is assigned an i.d. card which contains a unique 4 character (letter and digit (number)) barcode. Each barcode contains at most 1 digit. • How many unique i.d. cards are possible?

  10. Ex1: A.C. Principle • 0-digits (ALL letters) : There are no digits and 26 choices for each letter. • 26 x 26 x 26 x 26 • = 456,976 • 1-digit (1 num., 3 letters) : There are 10 choices for digits, 26 for letters • 10 x 26 x 26 x 26 • = 175,760 • The digit can be in any of the 4 positions, so 4 x 175,760 • = 703,040 • The last step is to ADD the two totals • 456,976 + 703,040 = 1,160,016

  11. Ex2: A.C. Principle • The combination for your gym locker consists of 4 symbols (letters and digits). If there are 1 or 0 letters, how many combinations are possible?

  12. Ex2: A.C. Principle • 0 letters: There are 10 digit choices • 10 x 10 x 10 x 10 • = 10,000 • 1 letter: There are 26 letter choices • 26 x 10 x 10 x 10 • = 26,000 • But remember that the letter can be in any position • So….. 4 x 26,000 • = 104,000 • ADD TOTALS: 10,000 + 104,000 = 114,000

  13. Ex 3. You Try: A.C. Principle • You are to create a code for your computer password. The code must consist of 3 symbols (letters and digits). How many password combinations are possible if at most one digit is used?

  14. Ex3: you try • 0 digits: • 26 x 26 x 26 • = 17,576 • 1 digit: • 10 x 26 x 26 • = 6,760 • DIGIT CAN BE IN ANY 3 POSITIONS • 6,760 X 3 = 20,280 • ADD • 20,280 + 6,760 = 27,040

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