Objectives

1. Objectives • Solve counting problems using the Multiplication Rule • Solve counting problems using permutations • Solve counting problems using combinations • Solve counting problems involving permutations with nondistinct items • Compute probabilities involving permutations and combinations

2. Do Now: • Write down all the items you would put in your fruit salad for the end of the year picnic :) • - New VOCABulary Combination Permutation Factorial • Practice Problems Block 2 • Midterm discussion & Update • Go over Tuesday’s HW • Objectives: • Solve problems using permutations • Solve problems using combinations

3. Precalculus 2!!! • over your midterm – are there questions you have or points you believe you deserve…specifically, look at your answers for permutations and combinations • DO NOW: Notes on the comparisons  • Portfolio discussion • QUIZ next week! • Objectives: • Define permutation, combination, factorial, favorable outcome • Solve probability, permutation and combination problems • HW: Portfolio!

4. Do Now: • Write down all the items you would put in your fruit salad for the end of the year picnic :) …order does not matter  How about your locker or bike lock combination? Does order matter??!! • Objectives: • Solve problems using permutations • Solve problems using combinations

5. Class election • Ok, so we have 16 students in the class and we are going to make a council of 3 student representatives. A pres, a vice pres and a treasure. • How many ways can we create our trio? • Does ‘order’ matter…?

6. Class election • Ok, so we have 16 students in the class and we are going to make a council of 3 student representatives of equal value? • How many ways can we create our trio? • Does ‘order’ matter…?

7. Vocabulary • Permutation – is an ordered arrangement in which r objects are chosen from n distinct (different) objects and repetition is not allowed. The symbol nPr represents the number of permutations of r objects selected from n objects. • Combination – is a collection, without regard to order, of n distinct objects without repetition. The symbol nCr represents the number of combinations of n objects taken r at a time..

8. Vocabulary • Permutation – is an ordered arrangement in which r objects are chosen from n distinct (different) objects and repetition is not allowed. The symbol nPr represents the number of permutations of r objects selected from n objects. • How would I represent the class election example?! • Combination – is a collection, without regard to order, of n distinct objects without repetition. The symbol nCr represents the number of combinations of n objects taken r at a time. • How would I represent the class election example?!

9. Vocabulary  • Fundamental Counting Principle • Use this when want to find the total ways (number) a task can occur. (license plate problem) Multiply! • (ABCDEF) = n(A) * n(B) *n(C) * n(D) * n(E) * n(F)

10. Vocabulary Handout  Ok, so in your own words define • Permutation and • Combination • Draw an example to help you remember… • -you can wait on rating “my understanding” until the end of class…

11. Permutations Number of Permutations of n Distinct Objects taken r at a time: N objects are distinct Once used an object cannot be repeated Order is important n! nPr = ----------- (n – r)! Factorial – n! is defined to be n! = n∙(n-1)∙(n-2)∙(n-3)….. (3)∙(2)∙(1)

12. Permutations Number of Permutations of n Distinct Objects taken r at a time: N objects are distinct Once used an object cannot be repeated Order is important n! nPr = ----------- (n – r)! How would I represent the class election example?!

13. Combinations Number of Combinations of n Distinct Objects taken r at a time: N objects are distinct Once used an object cannot be repeated (no repetition) Order is not important n! nCr = ----------- r!(n – r)! How would I represent the class election example?!

14. Combinations Number of Combinations of n Distinct Objects taken r at a time: N objects are distinct Once used an object cannot be repeated (no repetition) Order is not important n! nCr = ----------- r!(n – r)!

15. Combinations Number of Combinations of n Distinct Objects taken r at a time: N objects are distinct Once used an object cannot be repeated (no repetition) Order is not important n! nCr = ----------- r!(n – r)! r! in the denominator eliminates the double count!! In other words, ABC and BAC are the SAME in a combination! *in a permutation, they are different…think of pres, vp and treas…

16. How to Tell • Is a problem a permutation or a combination? • One way to tell • Write down one possible solution (i.e. Roger, Rick, Randy) • Switch the order of two of the elements (i.e. Rick, Roger, Randy) • Is this the same result? • If no – this is a permutation – order matters • If yes – this is a combination – order does not matter

17. Vocabulary Handout  Ok, so now how about your own example…you may put this is the further understanding box • And rate “my understanding” …1 is the least (amount of understanding )and 4 is the most

18. Your Turn… • Partner Up – pairs of 2 or groups of 3 and work on CW…I will be coming around to help and check  • We MISSED TWO DAYS AND I WAS IN A MEETING TUESDAY SO THIS IS OUR ONLY DAY THIS WEEK AND WE NEED TO STAY FOCUSED…grrrrreat 

19. Example 1 If there are 3 different colors of paint (red, blue, green) that can be used to paint 2 different types of toy cars (race car, police car), then how many different toys can there be?

20. Red Race Car Red Race Red Police Car Police Blue Race Car Blue Race Blue Police Car Police Green Race Car Green Race Green Police Car Police Example 1 Illustrated • A tree diagram of the different possibilities Paint Car Possibilities

21. Example 2 In a horse racing “Trifecta”, a gambler must pick which horse comes in first, which second, and which third. If there are 8 horses in the race, and every order of finish is equally likely, what is the chance that any ticket is a winning ticket? The probability that any one ticket is a winning ticket is 1 out of 8P3, or 1 out of 336

22. Permutations with replacement Number of Permutations of n Distinct Items taken r at a time with replacement: N objects are distinct Once used an object can be repeated (replacement) Order is important P = nr

23. Example 3 Suppose a computer requires 8 characters for a password. The first character must be a letter, but the remaining seven characters can be either a letter or a digit (0 thru 9). The password is not case-sensitive. How many passwords are possible on this computer? 26 • 367 = 2.037 x 1012

24. Example 4 If there are 8 researchers and 3 of them are to be chosen to go to a meeting, how many different groupings can be chosen?

25. Permutations – non-distinct items Number of Permutations with Non-distinct Items: N objects are not distinct K different groups n! P = --------------------- where n = n1 + n2 + … + nk n1!∙n2!∙ ….∙nk!

26. Example 5 How many different vertical arrangements are there of 9 flags if 4 are white, 3 are blue and 2 are red? 9! 9•8•7•6•5•4! 9•8•7•6•5 ----------- = ------------------ = --------------- = 1260 4!•3!•2! 4!•3!•2! 3•2•1•2•1

27. Permutation vs Combination • Comparing the description of a permutation with the description of a combination • The only difference is whether order matters

28. Multiplication Rule of Counting If a task consists of a sequence of choices in which there are p selections for the first item, q selections for the second item, and r choices for the third item, and so on, then the task of making these selections can be done in p ∙ q ∙ r ∙ ….. different ways • The classical method, when all outcomes are equally likely, involves counting the number of ways something can occur • This section includes techniques for counting the number of results in a series of choices, under several different scenarios

29. Vocabulary • Factorial – n! is defined to be n! = n∙(n-1)∙(n-2)∙(n-3)….. (3)∙(2)∙(1) • Permutation – is an ordered arrangement in which r objects are chosen from n distinct (different) objects and repetition is not allowed. The symbol nPr represents the number of permutations of r objects selected from n objects. • Combination – is a collection, without regard to order, of n distinct objects without repetition. The symbol nCr represents the number of combinations of n objects taken r at a time..

30. Summary and Homework • Summary • The Multiplication Rule counts the number of possible sequences of items • Permutations and combinations count the number of ways of arranging items, with permutations when the order matters and combinations when the order does not matter • Permutations and combinations are used to compute probabilities in the classical method • Homework