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Discrete Mathematics

Discrete Mathematics. Basics of Counting. University of Jazeera College of Information Technology & Design Khulood Ghazal. The product rule. If there are n 1 ways to do task 1, and n 2 ways to do task 2 Then there are n 1 n 2 ways to do both tasks in sequence

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Discrete Mathematics

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  1. Discrete Mathematics Basics of Counting University of Jazeera College of Information Technology & Design Khulood Ghazal

  2. The product rule • If there are n1 ways to do task 1, and n2 ways to do task 2 • Then there are n1n2 ways to do both tasks in sequence • This applies when doing the “procedure” is made up of separate tasks • We must make one choice AND a second choice

  3. Product rule example • There are 18 math majors and 325 CS majors • How many ways are there to pick one math major and one CS major? • Total is 18 * 325 = 5850

  4. The sum rule • If there are n1 ways to do task 1, and n2 ways to do task 2 • If these tasks can be done at the same time, then… • Then there are n1+n2 ways to do one of the two tasks • We must make one choice OR a second choice

  5. Sum rule example • There are 18 math majors and 325 CS majors • How many ways are there to pick one math major or one CS major? • Total is 18 + 325 = 343

  6. More complex counting problems • We combining the product rule and the sum rule • Thus we can solve more interesting and complex problems

  7. Example(1): • The chairs of an auditorium are 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?

  8. What is the largest number of chairs that can be labeled differently? • We can think of this problem as involving a sequence of two tasks: • Assign a letter between A and Z • Assign a number between 1 and 100 • The Product Rule says that there are 26 * 100 = 2600 ways to do this. • So we can label 2600 chairs.

  9. Example(2): • How many different license plates are available if each plate contains a sequence of three letters followed by three digits? • 26 choices for each letter • 10 choices for each digit • Total of: 26 * 26 * 26 * 10 * 10 * 10

  10. Example(3): • How many different bit strings are there of length seven? • You probably already know it is 27. • Think of this as: • 2 (2 (2 (2 (2 (2 * 2)))))

  11. Example(4): • How many different bit strings are there of length 1? Only 2: 0 or 1 • How many different bit strings are there of length 2? There are 4: 00, 01, 10, 11 • How many different bit strings are there of length 3? There are 8: 000, 001, 010, 011, 100, 101, 110, 111

  12. Example(5): • Each user on a computer system has a password • Each password is six to eight characters long • Each character is an uppercase letter or a digit • Each password must contain at least one digit • How many possible passwords are there?

  13. There are 26 letters and 10 decimal digits = 36 characters that we can use to form passwords. • For P6 (6-character) passwords, the Product Rule says there are 366 potential passwords. • But passwords that are all letters are prohibited. There are 266 of these. • So there are 366 – 266 for P6 passwords. • Similarly, for P7 and P8 passwords. • P7 = 367 – 267 • P8 = 368 – 268 • Total passwords = P6 + P7 + P8

  14. Example(6): • How many bit strings of length eight either start with a 1 or end with the two bits 00? • 1st Task: Construct a string beginning with a 1. • 2nd Task: Construct a string ending with 00. • Both tasks: Construct a string that begins with a 1 and ends with 00.

  15. 1st: There are 28 ways to construct a binary string of 8 bits, but it starts with a 1, so there are 27 ways to construct an 8-bit binary string starting with 1. • 2nd: Construct a string ending with 00. • The product rule says there are 2 ways to choose the first 6 bits and 1 way to chose the last 2 bits, so there are 26 ways to construct this string. • Both: Construct a string that begins with 1 and ends with 00. • The product rule says there is 1 way to choose the first bit, 2 ways to chose the middle 5 bits, and 1 way to chose the last 2 bits, so there are 25 ways to construct this string. • Total = (27 + 26) – 25 = 160

  16. Example(7): • A multiple choice test contains 10 questions. There are four possible answers for each question. • How many ways can a student answer the questions on the test if every question is answered? • 4*4*4*4*4*4*4*4*4*4 = 410 • How many ways can a student answer the questions on the test if the student can leave answers blank? • 5*5*5*5*5*5*5*5*5*5 = 510

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