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Counting Principles

Counting Principles. Multiplication rule Permutations Combinations. How many phone #’s can be generated in one area code if the first digit cannot be either 0 or 1? 8,000,000.

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Counting Principles

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  1. Counting Principles • Multiplication rule • Permutations • Combinations

  2. How many phone #’s can be generated in one area code if the first digit cannot be either 0 or 1? • 8,000,000

  3. High school faculty are to be issued special coded identification cards that consist of four letters of the alphabet.   How many different ID cards can be issued if the letters can be used more than once? How many if no letters can be repeated? • 456,976 358,800

  4. Permutations andCombinations

  5. In this section, techniques will be introduced for counting • the unordered selections of distinct objects and • the ordered arrangements of objects • of a finite set.

  6. Arrangements • The number of ways of arranging n unlike objects in a line is n !. • Note: n ! = n (n-1) (n-2) ···3 x 2 x 1

  7. Example • It is known that the password on a computer system contain • the three letters A, B and C • followed by the six digits 1, 2, 3, 4, 5, 6. • Find the number of possible passwords.

  8. Solution • There are 3! ways of arranging the letters A, B and C, and • 6! ways of arranging the digits 1, 2, 3, 4, 5, 6. • Therefore the total number of possible passwords is • 3! x 6! = 4320. • i.e. 4320 different passwords can be formed.

  9. Like Objects • The number of ways of arranging in a line • n objects, • of which p are alike, is

  10. The result can be extended as follows: • The number of ways of arranging in a line n objects • of which p of one type are alike, • q of a second type are alike, • r of a third type are alike, and so on, is

  11. Example • Find the number of ways that the letters of the word • STATISTICS • can be arranged.

  12. Solution • The word STATISTICS contains • 10 letters, in which • S occurs 3 times, • T occurs 3 times and • I occurs twice.

  13. Therefore the number of ways is • That is, there are 50400 ways of arranging the letter in the word STATISTICS.

  14. Example • A six-digit number is formed from the digits • 1, 1, 2, 2, 2, 5 and • repetitions are not allowed. • How many these six-digit numbers are divisible by 5?

  15. Solution • If the number is divisible by 5 then it must end with the digit • 5. • Therefore the number of these six-digit numbers which are divisible by 5 is equal to the number of ways of arranging the digits • 1, 1, 2, 2, 2.

  16. Then, the required number is • That is, there are 10 of these six-digit numbers are divisible by 5.

  17. Permutations • A permutation of a set of distinct objects is an orderedarrangement of these objects. • An ordered arrangement of r elements of a set is called an r-permutation. • The number of r-permutations of a set with n distinct elements,

  18. i.e. the number of permutations of r objects taken from n unlike objects is: • Note: 0! is defined to 1, so

  19. Example Find the number of ways of placing 3 of the letters A, B, C, D, E in 3 empty spaces.

  20. Solution • The first space can be filled in • 5 ways. • The second space can be filled in • 4 ways. • The third space can be filled in • 3 ways.

  21. Therefore there are • 5 x 4 x 3 ways • of arranging 3 letters taken from 5 letters. • This is the number of permutations of 3 objects taken from 5 and • it is written as P(5, 3), • so P(5, 3)= 5 x 4 x 3 = 60.

  22. On the other hand, 5 x 4 x 3 could be written as • Notice that the order in which the letters are arranged is important --- • ABC is a different permutation from ACB.

  23. Example • How many different ways are there to select • one chairman and • one vice chairman • from a class of 20 students.

  24. Solution • The answer is given by the number of 2-permutations of a set with 20 elements. • This is • P(20, 2) = 20 x 19 = 380

  25. Combinations • An r-combination of elements of a set is an unordered selection of r elements from the set. • Thus, an r-combination is simply a subset of the set with r elements.

  26. The number of r-combinations of a set with n elements, • where n is a positive integer and • r is an integer with 0 <= r <= n, • i.e. the number of combinations of r objects from n unlike objects is

  27. Example • How many different ways are there to select two class representatives from a class of 20 students?

  28. Solution • The answer is given by the number of 2-combinations of a set with 20 elements. • The number of such combinations is

  29. Example • A committee of 5 members is chosen at random from • 6 faculty members of the mathematics department and • 8 faculty members of the computer science department.

  30. In how many ways can the committee be chosen if • (a) there are no restrictions; • (b) there must be more faculty members of the computer science department than the faculty members of the mathematics department.

  31. Solution • (a) There are 14 members, from whom 5 are chosen. • The order in which they are chosen is not important. • So the number of ways of choosing the committee is • C(14, 5) = 2002.

  32. (b) If there are to be more • faculty members of the computer science department than • the faculty members of the mathematics department, • then the following conditions must be fulfilled.

  33. (i) 5 faculty members of the computer science department. • The number of ways of choosing is • C(8, 5) = 56. • (ii) 4 faculty members of the computer science department and • 1 faculty member of the mathematics department

  34. The number of ways of choosing is • C(8, 4) xC(6, 1) = 70 x 6 = 420. • (iii) 3 faculty members of the computer science department and • 2 faculty members of the mathematics department • The number of ways of choosing is • C(8, 3) xC(6, 2) = 56 x 15 = 840

  35. Therefore the total number of ways of choosing the committee is • 56 + 420 + 840 = 1316.

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