Chapter 10 Counting Methods

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”

Using the Fundamental Counting Principle Section 10-2 Using the Fundamental Counting Principle

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.

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.

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

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 Part of Task Select first digit Select second digit Number of ways 5 (0 can’t be used) 6 There are 5(6) = 30 two-digit numbers.

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 Part of Task Select first digit Select second digit Number of ways 5 (repeated digits not allowed) There are 5(5) = 25 two-digit numbers.

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

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!.

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

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

Definition of Zero Factorial

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

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

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.

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

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