1. SECTION 10-2 Using the Fundamental Counting Principle Slide 10-2-1

2. USING THE FUNDAMENTAL COUNTING PRINCIPLE Uniformity and the Fundamental Counting Principle Factorials Arrangements of Objects Slide 10-2-2

3. UNIFORMITY CRITERION FOR MULTIPLE-PART TASKS Slide 10-2-3 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.

4. FUNDAMENTAL COUNTING PRINCIPLE Slide 10-2-4 When a task consists of k separate parts and satisfies the uniformity criterion, if the first part can be done in n 1 ways, the second part can be done in n 2 ways, and so on through the k th part, which can be done in n k ways, then the total number of ways to complete the task is given by the product

5. EXAMPLE: TWO-DIGIT NUMBERS Slide 10-2-5 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 TaskSelect first digitSelect second digit Number of ways5 (0 can’t be used) 6 There are 5(6) = 30 two-digit numbers.

6. EXAMPLE: TWO-DIGIT NUMBERS WITH RESTRICTIONS Slide 10-2-6 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 65 (repeated digits not allowed) There are 6(5) = 30 two-digit numbers.

7. EXAMPLE: TWO-DIGIT NUMBERS WITH RESTRICTIONS Slide 10-2-7 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.

8. FACTORIALS Slide 10-2-8 For any counting number n, the product of all counting numbers from n down through 1 is called n factorial, and is denoted n!.

9. FACTORIAL FORMULA Slide 10-2-9 For any counting number n, the quantity n factorial is given by

10. EXAMPLE: Slide 10-2-10 Evaluate each expression. a) 4! b) (4 – 1)!c) Solution

11. DEFINITION OF ZERO FACTORIAL Slide 10-2-11

12. ARRANGEMENTS OF OBJECTS Slide 10-2-12 When finding the total number of ways to arrange a given number of distinct objects, we can use a factorial.

13. ARRANGEMENTS OF N DISTINCT OBJECTS Slide 10-2-13 The total number of different ways to arrange n distinct objects is n!.

14. EXAMPLE: ARRANGING BOOKS Slide 10-2-14 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.

15. ARRANGEMENTS OF N OBJECTS CONTAINING LOOK-ALIKES Slide 10-2-15 The number of distinguishable arrangements of n objects, where one or more subsets consist of look- alikes (say n 1 are of one kind, n 2 are of another kind, …, and n k are of yet another kind), is given by

16. EXAMPLE: DISTINGUISHABLE ARRANGEMENTS Slide 10-2-16 Determine the number of distinguishable arrangements of the letters of the word INITIALLY. Solution 9 letters total 3 I’s and 2 L’s