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Counting Techniques

Counting Techniques. Tree Diagram Multiplication Rule Permutations Combinations. Tree Diagram. a method of listing outcomes of an experiment consisting of a series of activities. Tree diagram for the experiment of tossing two coins. H. H. T. start. H. T. T.

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Counting Techniques

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  1. Counting Techniques • Tree Diagram • Multiplication Rule • Permutations • Combinations

  2. Tree Diagram a method of listing outcomes of an experiment consisting of a series of activities

  3. Tree diagram for the experiment of tossing two coins H H T start H T T

  4. Find the number of paths without constructing the tree diagram: Experiment of rolling two dice, one after the other and observing any of the six possible outcomes each time . Number of paths = 6 x 6 = 36

  5. Multiplication of Choices If there are n possible outcomes for event E1 and m possible outcomes for event E2, then there are n x m or nm possible outcomes for the series of events E1 followed by E2.

  6. Area Code Example Until a few years ago a three-digit area code was designed as follows. The first could be any digit from 2 through 9. The second digit could be only a 0 or 1. The last could be any digit. How many different such area codes were possible? 82 10 = 160

  7. Ordered Arrangements In how many different ways could four items be arranged in order from first to last? 4 3 21 = 24

  8. Factorial Notation • n! is read "n factorial" • n! is applied only when n is a whole number. • n! is a product of n with each positive counting number less than n

  9. Calculating Factorials 5! = 5 • 4 • 3 • 2 • 1 = 3! = 3 • 2 • 1 = 120 6

  10. Definitions 1! = 1 0! = 1

  11. Complete the Factorials: 24 4! = 10! = 6! = 15! = 3,628,800 720 1.3077 x 1012

  12. Permutations A permutation is an arrangement in a particular order of a group of items. There are to be no repetitions of items within a permutation.)

  13. Listing Permutations How many different permutations of the letters a, b, c are possible? Solution: There are six different permutations: abc, acb, bac, bca, cab, cba.

  14. Listing Permutations How many different two-letter permutations of the letters a, b, c, d are possible? Solution: There are twelve different permutations: ab, ac, ad, ba, ca, da, bc, bd, cb, db, cd, dc.

  15. Permutation Formula The number of ways to arrange in order n distinct objects, taking them r at a time, is:

  16. Another notation for permutations:

  17. Find P7, 3

  18. Applying the Permutation Formula P3, 3 = _______ P4, 2 = _______ P6, 2 = __________ P8, 3 = _______ P15, 2 = _______ 6 12 30 336 210

  19. Application of Permutations A teacher has chosen eight possible questions for an upcoming quiz. In how many different ways can five of these questions be chosen and arranged in order from #1 to #5? Solution: P8,5 = = 8• 7 • 6 • 5 • 4 = 6720

  20. Combinations A combination is a grouping in no particular order of items.

  21. Combination Formula The number of combinations of n objects taken r at a time is:

  22. Other notations for combinations:

  23. Find C9, 3

  24. Applying the Combination Formula 35 10 C5, 3 = ______ C7, 3 = ________ C3, 3 = ______ C15, 2 = ________ C6, 2 = ______ 105 1 15

  25. Application of Combinations A teacher has chosen eight possible questions for an upcoming quiz. In how many different ways can five of these questions be chosen if order makes no difference? Solution: C8,5 = = 56

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