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Discrete Mathematics Chapter 5 Counting 大葉大學 資訊工程系 黃鈴玲 酒店设计 Ch5-* ? A counting problem: (Example 15) Each user on a computer system has a password, which is six to eight characters long, where each characters is an uppercase letter or a digit. Each password must contain at least one digit. How many possible passwords are there? ? This section introduces a variety of other counting problems the basic techniques of counting. §5.1 The Basics of counting Ch5-* Basic counting principles ? The sum rule: If a first task can be done in n1 ways and a second task in n2 ways, and if these tasks cannot be done at the same time. then there are n1+n2 ways to do either task. Example 11 Suppose that either a member of faculty or a student is chosen as a representative to a university committee. How many different choices are there for this representative if there are 37 members of the faculty and 83 students? n1 n2 n1 + n2 ways Ch5-* Example 12 A student can choose a computer project from one of three lists. The three lists contain 23, 15 and 19 possible projects respectively. How many possible projects are there to choose from? Sol: 23+15+19=57 projects. ? The product rule: Suppose that a procedure can be broken down into two tasks. If there are n1 ways to do the first task and n2 ways to do the second task after the first task has been done, then there are n1 n2 ways to do the procedure. n1 n2 n1 × n2 ways Ch5-* Example 2 The chair of an auditorium (大禮堂) is 　to be labeled with a letter and a positive integer 　not exceeding 100. What is the largest number of 　chairs that can be labeled differently? Sol: 26 × 100 = 2600 ways to label chairs. letter Example 4 How many different bit strings are there of length seven? Sol:　1　2　 3　 4　 5　 6 　7 □ □ □ □ □ □ □ ↑ ↑ ↑ ↑ ↑ ↑ ↑ 0,1 0,1 0,1 . . . . . . 0,1 → 27 種 Ch5-* Example 5 How many different license plates (車牌) are avai