PreQuiz 12A Make a systematic list of all possible outcomes: You spin a spinner and flip a coin.

PreQuiz 12A AP(greater than 4) BP(prime) CP(not a 7) DP(less than 13) The number 1 is not a prime number

PreQuiz 12B In a bag there are 9 red marbles, 5 green marbles and 6 blue marbles. Find the following probabilities: with replacement without replacement a. P(green, blue) ________ ________ b. P(green, green) ________ ________ c. P(blue, green, red) ________ ____________

PreQuiz 12B A diner runs a special in which a person can choose from 5 entrees, 7 sodas, and 3 sides. How many different specials can a person choose.

Permutations and Combinations SECTION 12.3

Target 12.3 I can use permutations and combinations to compute probabilities of compound events and solve problems.

Suppose that the 14 people at this table decided to switch places. In how many ways can they arrange themselves?

_____ ______ _____ ______ _____ ______ _____ ______ _____ ______ _____ ______ _____ _____

_____ ______ _____ ______ _____ ______ _____ ______ _____ ______ _____ ______ _____ _____ 14·13·12·11·10·9·8·7·6·5·4·3·2·1

Factorial 14·13·12·11·10·9·8·7·6·5·4·3·2·1=14! n! - means to multiply all numbers from n to 1

The values of 1! through 4! are shown below: 1!=1 2!=1·2=2 3!=1·2·3=6 4!=1·2·3·4=24 Continue this list by finding the values of 5! through 13! If you do not make any mistakes, you should find that 13!=6,227,020,800

Example 1 In how many ways can 12 people be arranged for a photograph? ___ · ___ · ___ · ___· ___ · ___· ___ · ___· ___ · ___· ___ · ___

Permutation versus Combination 1. Picking a team captain, pitcher, and shortstop from a group. 1. Picking three team members from a group. 2. Picking your favorite two colors, in order, from a color brochure. 2. Picking two colors from a color brochure. 3. Picking first, second and third place winners. 3. Picking three winners.

Example 2 Twelve people need to be photographed, but there are only five chairs. (The rest of the people will be standing behind and their order does not matter.) How many ways can you sit the twelve people on the five chairs? ___ · ___ · ___ · ___· ___

Example 3 Eight horses-Alabaster, Beauty, Candy, Doughty, Excellente, Friday, Great One, and High 'n Mighty-run a race. In how many ways can the first, second, and third finishers turn out? ___ · ___ · ___

Example 3A Ten horses run a race. In how many ways can the first, second, and third finishers turn out? ___ · ___ · ___

When the order is important:  Making seating arrangements  Picking first, second, and third place winners  Lock combination  Letters in a word  Digits in a number New Name: Permutation Lock

Permutation an arrangement of things in a certain order. By definition, 0! = 1. Remember!

10 horses taken 3 at a time 10 P 3 10 horses taken 4 at a time 10 P 4 8 horses taken 3 at a time 8 P 3 8 horses taken 4 at a time 8 P 4 6 horses taken 3 at a time 11 horses taken 2 at a time

Example 3A Ten horses run a race. In how many ways can the first, second, and third finishers turn out? 10 horses taken 3 at a time

You try 8 P 3 = 8 P 4 = 12 P 4 = 5 P 2 =

When order is not important:  Picking team members  Choosing side dishes  Choosing pizza toppings  Picking photos for a collage

Example 4 Cafeteria workers plan to offer pizza with four different toppings: Pepperoni, mushrooms, sausage, and green pepper. If they always put two different toppings on each pizza, how many different pizzas can they make? The combination of 4 things takes 2 at a time

Combination a selection of things in any order.

Example 5 Your friend is having a party and has 15 games to choose from. There is enough time to play 4 games. In how many ways can you choose 4 games to play? The combination of 15 things takes 4 at a time

You try: 1. There are 20 members in a club. Five people are selected to go to the state conference. In how many ways can the five members be selected? 2. Your English teacher asked you to read 3 novels from a list of 10. In how many ways can you choose which books to read?

Evaluate each permutation or combination: 1. 7 P P P C C C 2

State if each scenario involves a permutation or a combination. Then find the number of possibilities: 7. The ski club with ten members is to choose three officers captain, co-captain & secretary, how many ways can those offices be filled? 8. Suppose you are asked to list, in order of preference, the three best movies you have seen this year. If you saw 12 movies during the year, in how many ways can the three best be chosen and ranked?

9. An election ballot asks voters to select three city commissioners from a group of six candidates. In how many ways can this be done? 10. To win the small county lottery, one must correctly select 3 numbers from 30 numbers. The order in which the selection is made does not matter. How many different selections are possible?