1. Counting Principles and Probability Digital Lesson

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The Fundamental Counting Principle states that if one event can occur m ways and a second event can occur n ways, the number of ways the two events can occur in sequence is m n. Definition of Fundamental Counting Principle 1 st Coin Tossed Start HeadsTails Heads Tails 2 nd Coin Tossed There are 2  2 different outcomes: {HH, HT, TH, TT}. Heads Tails 2 ways to flip the coin.

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Example:Fundamental Counting Principle Example: A meal consists of a main dish, a side dish, and a dessert. How many different meals can be selected if there are 4 main dishes, 2 side dishes and 5 desserts available? # of main dishes # of side dishes # of desserts 452  = There are 40 meals available. 40

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Definition of Permutation A permutation is an ordered arrangement of n different elements. How many permutations are possible using the three colors red, white, and blue? There are 3 choices for the first color, 2 choices for the second color and only 1 choice for the third color. 3! = 3 2 1 = 6 permutations “factorial”

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 __ x __ x __ Definition of n P r A permutation of n elements taken r at a time is a subset of the collection of elements where order is important. Five projects are entered in a science contest. In how many ways can the projects come in first, second, and third? 3 5 projects 54 5 4 3 = 60 ways 4 projects 3 projects 1 st 2 nd 3 rd

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Formula for n P r The formula for the number of permutations of n elements taken r at a time is # in the collection # taken from the collection

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 The T’s are not distinguishable. STATSSTATS Example: Distinguishable Permutation If some of the items are identical, distinguishable permutations must be used. In how many distinguishable ways can the letters STATS be written? STATS The S’s are not distinguishable. Example continues.

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Definition of Distinguishable Permutation The number of distinguishable permutations of the n objects is The letters STATS can be written in S’s T’s A’s where n = n 1 + n 2 + n 3 +... + n k.

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Definition of Combination A combination of n elements taken r at a time is a subset of the collection of elements where order is not important. Using the letters A, B, C, and D, find all the possible combinations using two of the letters. {AB} {AC} {AD} {BC} {BD} {CD} This is the same as {BA}. There are six different combinations using 2 of the 4 letters.

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Fornula for n C r The formula for the number of combinations of n elements taken r at a time is # in the collection # taken from the collection

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Example: Combination Example: How many different ways are there to choose 6 out of 10 books if the order does not matter? There are 210 ways to choose the 6 books. 3

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Graphing Utility: Permutation and Combination Graphing Utility: Permutation Graphing Utility: Combination