3. How many different ways can 10 students line up for lunch?

Warm up! • Christian is going to dinner at Olive Garden. He is allowed to choose one of four pastas, one of three sauces, and one of three meats. How many different pasta dished could Christian make? • 2. How would the number of combinations change if they were out of one of the sauces? 3. How many different ways can 10 students line up for lunch? 4. The student council of 32 members must choose a president, a vice president, a secretary, and a treasurer. How many combinations of officers could there be?

Let’s say there are 15 student council members named: Bruce, Shareeka, Yasmine, Jalen, Greg, Stephon, Mikey, Bria, Joseph, Ekure, Anthony, Ar-Keno, Brittany, Tiffany, and Amber. If Shareeka, Tiffany, Yasmine, and Bruce are elected, would the order in which they are chosen matter? IS… PresidentVice PresidentSecretaryTreasurer Shareeka Tiffany Yasmine Bruce the same as… Bruce Yasmine Tiffany Shareeka? 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 Notation

Permutation When deciding who goes 1st, 2nd, etc., order is important. A permutationis an arrangement or listing of objects in a specific order. The order of the arrangement is very important!! The notation for a permutation: n Pr = n is the total number of objects r is the number of objects selected (wanted) *Note if n = r then n Pr = n !

Permutations Simplify each expression. a. 12P2 b. 10P4 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

Permutation Example: Four runners are needed to run the 400 meter relay. How many different arrangements are there for the four runners to run the 4 legs of the race? The team has 8 sprinters. Four runners are needed to run the 400 meter relay. How many different arrangements are there for the four runners, choosing from the 8 team members, to run the 4 legs of the race? 24 1680

Practice: • Bugs Bunny, King Tut, Kevin Jerome, 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 Hamilton is picking a captain and co-captain from her 15 players. How many possibilities does she 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 a positive integer less thann, • 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?

Solution • The answer is given by the number of 2-combinations of a set with 20 elements. • The number of such combinations is

Example 2 From a class of 24, Mrs. Shaffer is randomly selecting 3 to help Mrs. Benson with a project. How many combinations are possible?

Your turn! For your school pictures, you can choose 4 backgrounds from a list of 10. How many combinations of backdrops are possible?

Your turn! Coach Lynch randomly selects 3 people out of his class of 20 to go help him get ready for a lacrosse match. How many possibilities of people does he have?

To Sum it Up: combinations permutations Both are counting principles that tell you the total number of possible outcomes "The combination to the safe is 472". "My fruit salad is a combination of apples, grapes and bananas" the order DOES matter the order doesn't matter A Permutation is an ordered Combination.

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 was 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, it is a Combination. • If the order does matter it is a Permutation. A Permutation is an ordered Combination.

Classwork Practice Worksheet How many ways???

Fill in the blanks…. A permutation is an _________________ of objects in which order ___________ matter. A combination is an _________________ of objects in which order ____________ matter.

homework • Pg 344 1-18 even • Pg 349 1-10 all, and 17 - 18

3. How many different ways can 10 students line up for lunch?