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Counting Methods: Fundamental Counting Principle and Factorials

Learn how to solve counting problems using the fundamental counting principle and factorials. Understand the uniformity criterion and apply it to multiple-part tasks. Examples and exercises provided.

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Counting Methods: Fundamental Counting Principle and Factorials

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  1. Chapter 10 Counting Methods

  2. Chapter 10: Counting Methods • 10.1 Counting by Systematic Listing • 10.2 Using the Fundamental Counting Principle • 10.3 Using Permutations and Combinations • 10.4 Using Pascal’s Triangle • 10.5 Counting Problems Involving “Not” and “Or”

  3. Section 10-2 • Using the Fundamental Counting Principle

  4. Using the Fundamental Counting Principle • Know the meaning of the uniformity in counting and understanding the fundamental counting principle. • Use the fundamental counting principle to solve counting problems. • Determine the factorial of a whole number. • Use factorials to determine the number of arrangements, including distinguishable arrangements, of a given number of objects.

  5. Uniformity Criterion for Multiple-Part Tasks A multiple-part task is said to satisfy the uniformity criterion if the number of choices for any particular part is the same no matter which choices were selected for the previous parts.

  6. Fundamental Counting Principle When a task consists of k separate parts and satisfies the uniformity criterion, if the first part can be done in n1 ways, the second part can be done in n2 ways, and so on through the k th part, which can be done in nk ways, then the total number of ways to complete the task is given by the product

  7. Example: Counting Two-Digit Numbers How many two-digit numbers can be made from the set {0, 1, 2, 3, 4, 5}? (numbers can’t start with 0.) Solution There are 5(6) = 30 two-digit numbers.

  8. Example: Building Two-Digit Numbers with Restrictions How many two-digit numbers that do not contain repeated digits can be made from the set {0, 1, 2, 3, 4, 5}? Solution There are 5(5) = 25 two-digit numbers.

  9. Example: Counting the Number of IDs How many ways can you select two letters followed by three digits for an ID? Solution There are 26(26)(10)(10)(10) = 676,000 IDs possible.

  10. Factorials For any counting number n, the product of all counting numbers from n down through 1 is called n factorial, and is denoted n!.

  11. Factorial Formula For any counting number n, the quantity n factorial is given by

  12. Example: Evaluate each expression. a) 4! b) (4 – 1)! c) Solution

  13. Definition of Zero Factorial

  14. Arrangements of Objects When finding the total number of ways to arrange a given number of distinct objects, we can use a factorial.

  15. Arrangements of n Distinct Objects The total number of different ways to arrange n distinct objects is n!.

  16. Example: Arranging Books How many ways can you line up 6 different books on a shelf? Solution The number of ways to arrange 6 distinct objects is 6! = 720.

  17. Arrangements of n Objects Containing Look-Alikes The number of distinguishable arrangements of n objects, where one or more subsets consist of look-alikes (say n1 are of one kind, n2 are of another kind, …, and nk are of yet another kind), is given by

  18. Example: Counting Distinguishable Arrangements Determine the number of distinguishable arrangements of the letters of the word INITIALLY. Solution 9 letters total 3 I’s and 2 L’s

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