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53 Fundamental counting principle 52 Factorials 51 Permutations 50 WP: Permutations 49 Combinations 48 WP: Combinations 53a Fundamental counting principle Fundamental Counting Principal = Fancy way of describing how one would determine the number of ways a sequence of events can take place.

53b Fundamental counting principle You are at your school cafeteria that allows you to choose a lunch meal from a set menu. You have two choices for the Main course (a hamburger or a pizza), Two choices of a drink (orange juice, apple juice) and Three choices of dessert (pie, ice cream, jello). 12 meals How many different meal combos can you select?_________ Method one: Tree diagram Lunch

Hamburger Apple Pie Icecream Jello Orange Pie Icecream Jello

Pizza Apple Pie Icecream Jello Orange Pie Icecream Jello 53c Fundamental counting principle

Method two: Multiply number of choices 2 x 2 x 3 = 12 meals Ex 2: No repetition During the Olympic 400m sprint, there are 6 runners. How many possible ways are there to award first, second, and third places? 1st 2nd 3rd

3 places ____ 6 x ____ 5 x ____ 4 = 120 different ways 53d Fundamental counting principle Ex 3: With repetition License Plates for cars are labeled with 3 letters followed by 3 digits. (In this case, digits refer to digits 0 - 9. How many possible plates are there? You can use the same number more than once.

___ 26 x ___ 26 x ___ 26 x ___ 10 x ___ 10 x ___ 10 = 17,576,000 plates Ex 4: Account numbers for Century Oil Company consist of five digits. If the first digit cannot be a 0 or 1, how many account numbers are possible? ___

8 x ___ 10 x ___ 10 x ___ 10 x ___ 10 = 80,000 different account #s 53e Fundamental counting principle We are going to collect data from cars in the student parking lot. License place 1 2

3 4 . . . . . 50 Vehicle color Factorials

Factorials You think math is hard? What about English? The bandage was wound around the wound. The farm was used to produce produce. The dump was so full that it had to refuse more refuse. 52a Factorials 5 4 3 2 1 = 5! Factorial 7!= 7 6 5 4 3 2 1 = 5040

7! 7 6 5 4 3 2 1 42 5! 5 4 3 2 1 8! 8 7 6 5 4 3 2 1 8 7 6 5!3! 5 4 3 2 1 3 2 1 3 2 1 8 7 56 1

51a Permutations Permutations = A listing in which order IS important. Can be written as: P(6,4) or P 6 4

P(6,4) Represents the number of ways 6 items can be taken 4 at a time.. Or 6 x 5 x 4 x 3 = 360 Find P(15,3) = _____ 2730 15 x 14 x 13 Or

6 (6-1) (6-2) (6-3) 51b Permutations - Activity Write the letters M A T H on the top of your paper. Compose a numbered list of different 4 letter Permutations. -(not necessarily words) On the bottom of your paper write how many different permutations you have come up with. Dont forget your Name, Date and Period before turning in. Hint: You may wish to devise a strategy or pattern for finding all of the permutations before you start.

50a WP: Permutations Use the same formula from section 52 to solve these WPs. Ex1. Ten people are entered in a race. If there are no ties, in how many ways can the first three places come out? 8 = 720 ___ 10 x ___ 9 x ___ Ex2. How many different arrangements can be made with letters in word

LUNCH? 4 the 3 x ___ 2 x ___ 1 = 120 5! or the ___ 5 x ___ x ___ Ex3. You and 8 friends go to a concert. How many different ways can you sit in the assigned seats? 9! = 362,880

50b WP: Permutations - Activity On a separate sheet of paper, use only the letters below to form as many words as possible. Dont forget Name, Date and Period. 1 2 3 4 . . . .

. 50 Mathematics Permutations Puzzle How long will a socalled Eight Day Clock run without winding? An ice cream parlor has five flavors of ice cream available. How many different twoscoop cones can be served?

49a Combinations Combinations = A listing in which order is NOT important. Can be written as: C(3,2) or C2 3

C(3,2) means the number of ways 3 items can be taken 2 at a time. (order does not matter) Ex. C(3,2) using the letters C A T CA CT AT Pr n Cr r! n n = total

r = What you want 49b Combinations Pr n Cr r! n C(7,2) n = total r = What you want

P2 7 C2 2! 7 7x6 2x1 =

42 2 = 21 Which is not an expression for the number of ways 3 items can be selected from 5 items when order is not considered? WP: Combinations If you were to take two apples

from three apples, how many would you have? 48a WP: Combinations Permutations = Order IS important P(8,3) = ___ 8 x ___ 7 x ___ 6 = 336 Combinations = Order does not matter

P (8,3) 336 336 56 C(8,3) = 3! 3 2 6 48b WP: Combinations

Ex1. A college has seven instructors qualified to teach a special computer lab course which requires two instructors to be present. How many different pairs of teachers could there be? C(7,2) = 76 21 2! Ex2. A panel of judges is to consist of six women and three men. A list of potential judges includes seven women and six

men. How many different panels could be created from this list? Women Men 6 5 4 20 C(7,6) C(6,3) = 7 6 5 4 3 2 7 6 5 4 3 2 1

3 2 1 7*20 = 140 140 choices WP: Perms and Combs Ex1. The school board has seven members The board must have three officers: a chairperson, an assistant chairperson, and a secretary. a. How many different sets of these officers can be formed from this board? b. How many three-person committees can be formed from this board?

Is part (a) asking for a number of permutaions or a number of combinations? What about part (b)? How are your answers to parts (a) and (b) related? WP: Perms and Combs Ex2. Ralph has room for three plants on a windowsill. a. In how many different ways can three plants be arranged on his windowsill? b. Was (a) a permutation or a combination? c. Suppose Ralph has six plants. How many groups of three plants can be put on his windowsill? d. Was (c) a permutation or a combination?

e. Suppose Ralph has nine plants. How many ways can three of these plants be arranged on his windowsill? f. Was (e) a permutation or a combination? WP: Perms and Combs Ex3. To open your locker, you must dial a sequence of three numbers called the locks combination. Given that there are 40 numbers on a lock, how many different locker combinations are there?