4.4: The Counting Rules

Transcription

1 4.4: The Counting Rules The counting rules can be used to discover the number of possible for a sequence of events. Fundamental Counting Rule In a sequence of n events in which the first one has k 1 possibilities and the second has k 2 and the third has k 3 and so forth, the total number of possibilities of the sequence will be k 1 *k 2 *k 3...k n

2 Tossing a Coin and Rolling a Die A coin is tossed and a die is rolled. Find the number of possible outcomes. 1 heads tails Examples If you toss 9 coins, how many outcomes will you have? If you roll the die 4 times, how many outcomes will you have?

3 Types of Paint A paint manufacturer wishes to manufacture several different paints. The categories include: Color: red, blue, black,green, brown, yellow Type: Latex,oil Texture: flat, semigloss, high gloss Use: outdoor indoor How many different options do you have if you can pick one color, one type, one texture and one use? Distribution of Blood Types There are 4 blood types: A,B,AB and O. Blood can also be Rh+ and Rh. Finally, a blood donor can be classified as either male or female. How many different ways can a donor have his or her blood labeled?

4 When determining the number of different possibilities of a sequence of events, you must know whether repetitions are permissible. A store manager wants to make a four digit identification card for her employees. How many different cards can be made if she only used the digits 1,2,3,4,5 and 6 and repetitions are permitted? What if repetitions are not permitted? Factorial Notation! is the symbol for factorial n! = n*(n 1)*(n 2)...*1 0! = 1 4! = 8! =

5 Business Location Suppose a business owner has a choice of 5 locations in which to establish her business. She decides to rank each location according to certain criteria such as price and parking facilities. How many different ways can she rank the 5 locations? Business Location Suppose she wishes to only rank the top 3 of the 5 locations. How many different ways can she rank them?

6 Permutation A permutation is an arrangement of n objects in a specific order. n! is a permutation Permutation Rule The arrangement of n objects in a specific order using r objects at a time is called a permutation of n objects taking r objects at a time. It is written as n P r and the formula is: np r = n! (n r)! 6P 4 = 6! (6 4)!

7 Television News Stories A television news director wishes to use 3 news stories on an evening show. One story will be the lead story, one will be the second story, and the last will be a closing story. If the director has a total of 8 stories to choose from, how many possible ways can the program be set up? School Musical Plays A school musical director can select 2 musical plays to present next year. One will be presented in the fall, and one will be presented in the spring. If she has 9 to pick from, how many different possibilities are there?

8 Combinations Suppose a dress designer wishes to select two colors of material to design a new dress and she has on hand four colors. How many different possibilities can there be in this situation? Combinations A selection of distinct objects without regard to order is called a combination (Permutation order matters)

9 Letters Given the letters A,B,C,D, list the permutations and then combinations for selecting two letters. Letters Given the letters A,B,C,D, list the permutations and then combinations for selecting two letters.

10 Combination Rule The number of combinations of r objects selected from n objects is denoted by n C r and is given by the formula nc r = n! (n r)!r! Notice that n C r is the same as the formula for n P r with r! in the denominator. So we can write the formula for combinations as: nc r = np r r! The r! divides out all of the duplicates! (There are r! duplicates in all permutations/ combinations)

11 Book Reviews A newspaper editor has received 8 books to review. He decides that he can use 3 reviews in his newspaper. How many different ways can these 3 reviews be selected? Committed Selection In a club of 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. How many different possibilities are there? (Hint: picking men and women are two separate events)

12 Summary Fundamental Counting Rule: number of ways of a sequence of n events can occur if the first event can occur k 1 ways, the second k 2 etc. Permutation Rule: number of outcomes given n objects and choosing r objects (order matters) Combination Rule: number of outcomes given n objects and choosing r objects (order doesn't matter)