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1. Hey, Look! It’s a powerpoint! SWEET!

2. Permutations & Combinations Today’s Question: What is the difference between a permutation and combination? Standard: MM1D1.b.

3. 7 1 • • 6 • 5 • 4 • 3 • 2 • 1. Many mp3 players can vary the order in which songs are played. Your mp3 currently only contains 8 songs. Find the number of orders in which the songs can be played. There are 40,320 possible song orders. 1st Song2nd3rd4th5th6th7th8thOutcomes In this situation it makes more sense to use the Fundamental Counting Principle. = 8 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.

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

5. Factorial P E/F M/D A/S 2.) Simplify each expression. • 4! • 6! • (3²-4)!-119 d. 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 • 4 • 3 • 2 • 1)-119 = 1 = 5! = 5 • 4 • 3 • 2 • 1 = 120

6. 3. 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. PresidentViceSecretary TreasurerOutcomes In this situation it makes more sense to use the Fundamental Counting Principle. = 12 • 15 • • 13 14 32,760

7. Let’s say the student council members’ names were: Hunter, Bethany, Justin, Madison, Kelsey, Mimi, Taylor, Grace, Maighan, Tori, Alex, Paul, Whitney, Randi, and Dalton. If Hunter, Maighan, Whitney, and Alex are elected, would the order in which they are chosen matter? IS… PresidentVice PresidentSecretaryTreasurer Hunter Maighan Whitney Alex the same as… Whitney Hunter Alex Maighan ? 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.

8. Permutation Notation

9. 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:   nPr = n  is the total number of objects r is the number of objects selected (wanted) *Note  if  n = r   then   nPr  =  n!

10. Permutations nPr = 4.) 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

11. On your Own Permutation Examples: 5.) 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? 6.) A race has 8 athletes, but only 4 will win points for their team. In how many ways can the positions be won? 24

12. Rearrangements: 7.) Find the number of ways you can arrange all the letters in the word NEAT. Find the number of ways you can arrange 2 of the letters in the word NEAT.

13. Find Probability 8.) There are 10 players on a softball team. Each game the batting order is chosen randomly. Find the probability that you are chosen first, and your best friend is chosen to bat second. P(A)=Favorable Outcome Total Outcomes (Choices)

14. Combinations • A selection of objects in which order is not important. • Example 1 – 8 people pair up to do an assignment. How many different pairs are there?

15. 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

16. Combinations • the number of combinations of r objects from n unlike objects is • Example 1: 8 people pair up to do an assignment. How many different pairs are there?

17. Example 2 • How many different ways are there to select two class representatives from a class of 20 students?

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

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

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

21. 5.)Your turn! Coach randomly selects 3 people out of his class of 20 to go to the courts and help him get ready for a tennis match. How many possibilities of people does he have?

22. 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.

23. 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.

24. 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.

25. 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.