UNIT 4: Applications of Probability UNIT QUESTION: How do you calculate the probability of an event? Today’s Question: What is the difference between a permutation and a combination?

Use an appropriate method to find the number of outcomes in each of the following situations: 1. Your school cafeteria offers chicken or tuna sandwiches; chips or fruit; and milk, apple juice, or orange juice. If you purchase one sandwich, one side item and one drink, how many different lunches can you choose? There are 12 possible lunches. Sandwich(2) Side Item(2) Drink(3) Outcomes apple juice orange juice milk chicken, chips, apple chickn, chips, orange chicken, chips, milk chicken, fruit, apple chicken, fruit, orange chicken, fruit, milk tuna, chips, apple tuna, chips, orange tuna, chips, milk tuna, fruit, apple tuna, fruit, orange tuna, fruit, milk chips fruit chicken tuna chips fruit

Multiplication Counting Principle Ex 2: Georgia, the standard license plate has 3 letters followed by 4 digits. How many standard license plates are available in Georgia? 26 * 26 * 26 * 10 * 10 * 10 * 10 175,760,000

Multiplication Counting Principle Ex. 3: At Harrison, the locks on the lockers have numbers 0-49. Each combination uses 3 numbers. How many locker combinations are possible (assuming you can repeat numbers)? * * 50 50 50 125,000

You don’t want to play the same song twice! Ex 4: Your iPod players can vary the order in which songs are played. Your iPod currently only contains 8 songs (if you’re a loser). Find the number of orders in which the songs can be played. There are 40,320 possible song orders. 1st Song 2nd 3rd 4th 5th 6th 7th 8th Outcomes You don’t want to play the same song twice! 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320 The solution in this example involves the product of all the integers from n to one (n is representing the starting value). The product of all positive integers less than or equal to a number is a factorial.

Factorial The product of counting numbers beginning at n and counting backward to 1 is written n! and it’s called n factorial. factorial. EXAMPLE with Songs ‘eight factorial’ 8! = 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320

Factorial Simplify each expression. 4! 6! 4 • 3 • 2 • 1 = 24 c. For the 8th grade field events there are five teams: Red, Orange, Blue, Green, and Yellow. Each team chooses a runner for lanes one through 5. Find the number of ways to arrange the runners. 4 • 3 • 2 • 1 = 24 6 • 5 • 4 • 3 • 2 • 1 = 720 = 5! = 5 • 4 • 3 • 2 • 1 = 120

Ex 5: The student council of 15 members must choose a president, a vice president, a secretary, and a treasurer. There are 32,760 permutations for choosing the class officers. President Vice Secretary Treasurer Outcomes In this situation it makes more sense to use the Fundamental Counting Principle. = 15 • 14 • 13 • 12 32,760

Is Hunter Meghan Whitney Alex Let’s say the student council members’ names were: Hunter, Bethany, Justin, Madison, Kelsey, Mimi, Taylor, Grace, Meghan, Tori, Alex, Paul, Whitney, Randi, and Dalton. If Hunter, Maighan, Whitney, and Alex are elected, would the order in which they are chosen matter? President Vice President Secretary Treasurer Is Hunter Meghan Whitney Alex the same as… Whitney Hunter Alex Meghan? Although the same individual students are listed in each example above, the listings are not the same. Each listing indicates a different student holding each office. Therefore we must conclude that the order in which they are chosen matters.

Permutation When deciding who goes 1st, 2nd, etc., order is important. A permutation  is an arrangement or listing of objects in a specific order.  The order of the arrangement is very important!!  nPr = *Note  if  n = r   then   nPr  =  n!

Permutation Notation The number of arrangements of “n” objects, taken “r” at a time.

Permutations b. 10P4 Simplify each expression. a. 12P2 12 • 11 = 132 c. At a school science fair, ribbons are given for first, second, third, and fourth place, There are 20 exhibits in the fair. How many different arrangements of four winning exhibits are possible? 12 • 11 = 132 10 • 9 • 8 • 7 = 5,040 = 20P4 = 20 • 19 • 18 • 17 = 116,280

Permutations Bugs Bunny, King Tut, Mickey Mouse and Daffy Duck are going to the movies (they are best friends). How many different ways can they sit in seats A, B, C, and D below? 2. Coach is picking a captain and co-captain from 15 seniors. How many possibilities does he have if they are all equally likely? 24 A B C D 210

Combinations A selection of objects in which order is not important. Example – 8 people pair up to do an assignment. How many different pairs are there?

Combinations AB AC AD AE AF AG AH BA BC BD BE BF BG BH CA CB CD CE CF CG CH DA DB DC DE DF DG DH EA EB EC ED EF EG EH FA FB FC FD FE FG FH GA GB GC GD GE GF GH HA HB HC HD HE HF HG

Combinations The number of r-combinations of a set with n elements, where n is a positive integer and r is an integer with 0 <= r <= n, i.e. the number of combinations of r objects from n unlike objects is

Example 1 How many different ways are there to select two class representatives from a class of 20 students? The number of such combinations is:

Example 2 From a class of 24, the teacher is randomly selecting 3 to help Mr. Griggers with a project. How many combinations are possible?

You try: For your school pictures, you can choose 4 backgrounds from a list of 10. How many combinations of backdrops are possible?

Clarification on Combinations and Permutations "My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.

Clarification on Combinations and Permutations "The combination to the safe is 472". Now we do care about the order. "724" would not work, nor would "247". It has to be exactly 4-7-2.

To sum it up… If the order doesn't matter, then it is a Combination. If the order does matter, then it is a Permutation. A Permutation is an ordered Combination.

Classwork p. 178, #1-8

Homework p. 178-180, #9-15 all, 17-31 odd, 35-43 odd