The document discusses fundamental counting principles for determining the number of possible outcomes of independent events. It covers permutations, or arrangements that consider order, and combinations, which do not consider order. Examples include determining the number of possible outfits, license plates, and committee selections using formulas for permutations and combinations.

1. Probability

2. Objectives: Apply fundamental counting principle Compute permutations Compute combinations

3. Fundamental Counting Principle With repetition Without repetition

4. Fundamental Counting Principle Fundamental Counting Principle can be used to determine the number of possible outcomes when there are two or more characteristics. Fundamental Counting Principle states that if an event has m possible outcomes and another independent event has n possible outcomes, then there are m * n possible outcomes for the two events together.

5. Fundamental Counting Principle Lets start with a simple example. For a college interview, Robert has to choose what to wear from the following: 4 slacks, 3 shirts, 2 shoes and 5 ties. How many possible outfits does he have to choose from? 4*3*2*5 = 120 outfits

6. Fundamental Counting Principle At a restaurant at Cedar Point, you have the choice of 8 different entrees, 2 different salads, 12 different drinks, & 6 different desserts. How many different dinners (one choice of each) can you choose? 8*2*12*6 = 1152 different dinners

7. Fundamental Counting Principle with Repetition Ohio Licenses plates have 3 #’s followed by 3 letters. A. How many different licenses plates are possible if digits and letters can be repeated? 10*10*10*26*26*26 = 17,576,000 different plates

8. Fundamental Counting Principle without Repetition B. How many plates are possible if digits and numbers cannot be repeated? 10*9*8*26*25*24 = 11,232,000 plates

9. Fundamental Counting Principle How many different 7 digit phone numbers are possible if the 1 st digit cannot be a 0 or 1? 8*10*10*10*10*10*10 = 8,000,000 different numbers

10. Practice Problems Get ½ Sheet of Pad Paper (Crosswise)

11. Police use photographs of various facial features to help witnesses identify suspects. One basic identification kit contains 195 hairlines, 99 eyes and eyebrows, 89 noses, 105 mouths, and 74 chins and cheeks. The developer of the identification kit claims that it can produce billions of different faces. Is this claim correct? Determine how many different license plates are possible in 2 digits followed by 4 letters if (a) digits and letters can be repeated (b) digits and letters cannot be repeated

12. Permutation With repetition Without repetition

13. Permutations A Permutation is an arrangement of items in a particular order. Notice, ORDER MATTERS! To find the number of Permutations of n items, we can use the Fundamental Counting Principle or factorial notation .

14. Finding Permutations of n Objects Taken r at a Time To find the number of Permutations of n items chosen r at a time, you can use the formula

15. Permutations of n Objects Taken r at a Time Find the number of ways to arrange 6 items in groups of 4 at a time where order matters. Example 1

16. From a club of 24 members, a President, Vice President, Secretary, Treasurer and Historian are to be elected. In how many ways can the offices be filled? Example 2 Permutations of n Objects Taken r at a Time

17. Finding Permutations with Repetition The number of distinguishable permutations of n objects where one object is repeated q 1 times, another is repeated q 2 times, and so on is:

18. Find the number of distinguishable permutations of the letters in a) OHIO and b) MISSISSIPPI. Example 1A a) OHIO Finding Permutations with Repetition

19. Find the number of distinguishable permutations of the letters in a) OHIO and b) MISSISSIPPI. Example 1B b) MISSISSIPPI Finding Permutations with Repetition

20. Practice Problems Get ½ Sheet of Pad Paper (Crosswise)

21. How many 3 letter words can we make with the letters in the word LOVE? What is the total number of possible 5-letter arrangements of the letters w, h, i, t, e, if each letter is used only once in each arrangement?

22. Combination

23. Combination A Combination is an arrangement of r objects, WITHOUT regard to ORDER and without repetition, selected from n distinct objects is called a combination of n objects taken r at a time.

24. Find the number of ways to take 4 people and place them in groups of 3 at a time where order does not matter. Example 1 Combination

25. You are going to draw 4 cards from a standard deck of 52 cards. How many different 4 card hands are possible? . Example 2 Combination

26. Practice Problems Get ½ Sheet of Pad Paper (Crosswise)

27. In how many ways can you select a committee of 3 people from a group of 12 members? A man has, in his pocket, a silver dollar, a half-dollar, a quarter, a dime, a nickel, and a penny. If he reaches into his pocket and pulls out three coins, how many different sums may he have?

28. More Problems

29. In how many ways can you list your 3 favourite desserts, in order, from a menu of 10? A women has 4 blouses, 3 skirts, and 5 pairs of shoes. Assuming the woman does not care what she looks like, how many different outfits can she wear? Jack is the Chairman of a committee. In how many ways can a committee of 5 be chosen from 8 people given that Jack must be one of them? In how many ways can you select a committee of 3 people from a group of 12 members? The committee members consist of a chairperson, treasurer and a secretary.