8 - ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W L...

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ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c ° W W L Chen, X T Duong and Macquarie University, 1999. This work is available free, in the hope that it will be useful. Any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, with or without permission from the authors. Chapter 8 ELEMENTARY COUNTING TECHNIQUES 8.1. The Fundemental Principle of Counting We begin by studying two very simple examples. Example 8.1.1. Consider the collection of all 2-digit numbers where the Frst digit is either 1 or 2, and where the second digit is either 6, 7 or 8. Clearly there are 6 such numbers and they are listed below: 16 17 18 26 27 28 Arranged this way, we note that each row corresponds to a choice for the Frst digit and each column corresponds to a choice for the second digit. We have 2 rows and 3 columns, and hence 2 × 3=6 possibilities. Example 8.1.2. Consider the collection of all 3-digit numbers where the Frst digit is either 1, 2, 3 or 4, where the second digit is either 5 or 6, and where the third digit is either 7, 8 or 9. The candidates are listed below: 157 158 159 167 168 169 257 258 259 267 268 269 357 358 359 367 368 369 457 458 459 467 468 469 Arranged this way, we note that each block corresponds to a choice for the Frst digit. Within each block, each row corresponds to a choice for the second digit and each column corresponds to a choice for the third digit. We have 4 blocks, each with 2 rows and 3 columns, and hence 4 × 2 × 3 = 24 possibilities. These two examples are instances of a simple but very useful principle. This chapter was written at Macquarie University in 1999.

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8–2 W W L Chen and X T Duong : Elementary Mathematics FUNDAMENTAL PRINCIPLE OF COUNTING. Suppose that a frst event can occur in n 1 diFerent ways, a second event can occur in n 2 diFerent ways, and so on, and a k -th event can occur in n k diFerent ways. Then the number o± diFerent ways ±or these k events to occur in succession is given by n 1 × n 2 × ... × n k . Example 8.1.3. Consider motor vehicle licence plates made up of 3 letters followed by 3 digits, such as ABC012 . To determine the total number of possible diFerent licence plates, note that there are 26 choices for each letter and 10 choices for each digit. Hence the total number is 26 × 26 × 26 × 10 × 10 × 10. On the other hand, if the ±rst digit is restricted to be non-zero, then the total number is only 26 × 26 × 26 × 9 × 10 × 10. ²urthermore, if the letters are required to be distinct and the ±rst digit is restricted to be non-zero, then the total number is only 26 × 25 × 24 × 9 × 10 × 10. ²inally, if the letters and digits are required to be distinct and the ±rst digit is restricted to be non-zero, then the total number is only 26 × 25 × 24 × 9 × 9 × 8.
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This note was uploaded on 10/19/2010 for the course MATHEMATIC Math123 taught by Professor Goh during the Spring '10 term at UCLA.

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8 - ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W L...

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