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 start H T H 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 E 1 and m possible outcomes for event E 2, then there are n x m or nm possible outcomes for the series of events E 1 followed by E 2.

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: 4! = 10! = 6! = 15! = 24 3,628,800 720 1.3077 x 10 12

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 P 7, 3

18. Applying the Permutation Formula P 3, 3 = _______ P 4, 2 = _______ P 6, 2 = __________ P 8, 3 = _______ P 15, 2 = _______ 612 30336 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: P 8,5 = = 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 C 9, 3

24. Applying the Combination Formula C 5, 3 = ______ C 7, 3 = ________ C 3, 3 = ______ C 15, 2 = ________ C 6, 2 = ______ 35 1 105 10 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: C 8,5 = = 56