SECTION 10-2 Using the Fundamental Counting Principle Slide

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

UNIFORMITY CRITERION FOR MULTIPLE-PART TASKS Slide 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 Slide 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

EXAMPLE: TWO-DIGIT NUMBERS Slide 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.

EXAMPLE: TWO-DIGIT NUMBERS WITH RESTRICTIONS Slide 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.

EXAMPLE: TWO-DIGIT NUMBERS WITH RESTRICTIONS Slide 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 There are 26(26)(10)(10)(10) = 676,000 IDs possible.

FACTORIALS Slide 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 Slide For any counting number n, the quantity n factorial is given by

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

DEFINITION OF ZERO FACTORIAL Slide

ARRANGEMENTS OF OBJECTS Slide 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 Slide The total number of different ways to arrange n distinct objects is n!.

EXAMPLE: ARRANGING BOOKS Slide 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 Slide 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

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