The document discusses various counting principles including the fundamental counting principle, permutations, combinations, and probabilities. It provides examples of how to use these principles to calculate the number of possible outcomes in situations like choosing options, arranging objects in order, and selecting objects without regard to order.

1. Counting Principles Chapter 3.4

2. Objectives Use the Fundamental Counting Principle to find the number of ways 2 or more events can occur Find the number of ways a group of objects can be arranged in order Find the number of ways to choose several objects from a group without regard to order Use counting principles to find probabilities

3. The Fundamental Counting Principle If 1 event can occur in m ways and a second event can occur in n ways, the number of ways the 2 events can occur in sequence is m*n. This rule can be extended for any number of events occurring in sequence.

4. Buying a car You can choose a Ford, Subaru, or Porsche You can choose a small or medium car You can choose red (R), purple (P), white(W) or green (G) How many choices do you have? 3*2*3 = 18 choices

5. You can draw a tree diagram to show this

6. Security Systems The access code for a car’s security system consists of 4 digits. Each digit can be 0 through 9. How many access codes are possible if each digit can be used only once and not repeated? 10*9*8*7 = 5040

7. Security Systems The access code for a car’s security system consists of 4 digits. Each digit can be 0 through 9. How many access codes are possible if each digit can be repeated? 10*10*10*10 = 10,000

8. License Plates How many license plates can you make if a license plate consists of Six (out of 26) letters each of which can be repeated? Six (out of 26) letters each of which can not be repeated?

9. Permutations A permutation is an ordered arrangement of objects. The number of different permutations of n distinct objects is n! n! is read “n factorial” 5! = 5*4*3*2*1 0! is defined as 1

10. Baseball example How many starting lineups are possible for a 9 player baseball team? 9*8*7*6*5*4*3*2*1 = 362,880 Where is the factorial key on your calculator?

11. Finding n P r Find the number of ways of forming 3-digit codes in which no digit is repeated: Select the first digit (10 choices) Select the second digit (9 choices for each of the possible 10 first choices = 90) Select the third (8 choices for each of the 90 previous choices = 720)

12. Permutations of n Objects taken r at a Time The number of permutations of n distinct objects taken r at a time is n P r = n! , where r ≤ n (n-r)! Read this “Probability of n choose r”

13. Permutations of n Objects taken r at a Time Use the formula for the last problem n P r = n! , where r ≤ n (n-r)! 10 P 3 = 10! = 10*9*8*7*6*5*4*3*2*1 (10-3)! 7*6*5*4*3*2*1

14. Permutations of n Objects taken r at a Time

15. Another example . . .

16. AAAB How many ways can this be arranged? AAAB AABA ABAA BAAA 4

17. AAAABBC How many ways can this be arranged? 105, but it would be tedious writing these all out, so let’s use a formula:

18. Distinguishable Permutations

19. Try it with AAAB: 4! = 4*3*2*1 = 4 3!*1! 3*2*1

20. Try it with AAAABBC 7! = 7*6*5 = 105 4!*2!*1! 2

21. Combinations You want to buy 3 hats from a selection of 5. How many possible choices do you have? ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE 10 Choices

22. Combinations A combination is a selection of r objects from a group of n objects without regard to order and is denoted by n C r. The number of combinations of r objects selected from a group of n objects is n C r = = n! . (n-r)!r!

23. Combinations Try this with the 5 hats, choose 3: n C r = = n! . (n-r)!r! n C r = 5! . = 5*4*3*2*1 = 10 (5-3)!*3! 2*1*3*2*1

24. MISSISSIPPI A word consists of 1 M, 4 I’s, 4 S’s, and 2 P’s. If the letters are randomly arranged in order, what is the probability that the arrangement spells out Mississippi? How many distinguishable permutations? 11! = 1!*4!*3!*2! 34,650 Probability(Mississippi) = 1/34,650 = .000029

25. Diamond Flush Find the probability of being dealt 5 diamonds from a standard deck of cards. How many ways to choose 5 out of 13 diamonds? 13 C 5 How many ways to choose a 5 card hand? 52 C 5 P(diamond flush) = 13 C 5 = 1,287 = .0005 52 C 5 = 2,598,960

26. Homework P. 157 12-40 every 4 th problem

27. Now try this . . . With Skittles!