11.7 The Fundamental Counting Principle and Permutations 721 The fundamental counting principle is illustrated in Examples 1, 2, and 3. Why This Is Important The number of ordered arrangements of a set of objects is important to many applications including computer passwords, zip codes, telephone numbers, and Social Security numbers. Probability related to such arrangements is important to many branches of mathematics, science, engineering, and business. The Fundamental Counting Principle In Section 11.4, we introduced the fundamental counting principle, which is repeated here for your convenience. Definition: Fundamental Counting Principle If a first experiment has M distinct outcomes and a second experiment has N distinct outcomes, then the two experiments in that specific order have M N⋅ distinct outcomes. Example 1 License Plate Numbers A license plate “number” consists of two uppercase letters followed by four digits. Determine how many different license plate numbers are possible if a) repetition of letters and digits is permitted. b) repetition of letters and digits is not permitted. c) the first letter must be a vowel A E I O U ( , , , , ) and the first digit cannot be a 0, and repetition of letters and digits is not permitted. Solution There are 26 letters and 10 digits (0–9). We have six positions to fill, as indicated. LLDDDD a) Since repetition is permitted, there are 26 possible choices for both the first and second positions. There are 10 possible choices for the third, fourth, fifth, and sixth positions. 26 L 26 L 10 D 10 D 10 D 10 D Since 26 26 10 10 10 10 6,760,000, ⋅ ⋅ ⋅ ⋅ ⋅ = there are 6,760,000 different possible arrangements. b) There are 26 possibilities for the first position. Since repetition of letters is not permitted, there are only 25 possibilities for the second position. The same reasoning is used when determining the number of digits for positions 3 through 6. 26 L 25 L 10 D 9 D 8 D 7 D Since 26 25 10 9 8 7 3,276,000, ⋅ ⋅ ⋅ ⋅ ⋅ = there are 3,276,000 different possible arrangements. c) Since the first letter must be an A E I O , ,, ,or U, there are five possible choices for the first position. The second position can be filled by any of the letters except for the vowel selected for the first position. Therefore, there are 25 possibilities for the second position. Since the first digit cannot be a 0, there are nine possibilities for the third position. The fourth position can be filled by any digit except the one selected Learning Catalytics Keyword: Angel-SOM-11.7 (See Preface for additional details.) Copyright © 2022 by Jef Poskanzer <jef@ mail.acme.com>. All rights reserved.
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