Fundamental Principles of Counting, Combinations, Permutations Section 2.3 Collectively known as Combinatorics. Fundamental Principle of Counting: Construct a tree diagram to keep track of all possibilities. Each decision made produces new branches. Example 1: From a menu that offers 3 salads, 2 entrees, 3 desserts; how many different combinations of dinner can be made? Dessert 1 OR: Entrée 1 Dessert 2 Dessert 3 3 X 2 X 3 = 18 Salad 1 Dessert 1 Decision 1: 3 choices Entrée 2 Dessert 2 Decision 2: 2 choices Dessert 3 Decision 3: 3 choices
Dessert 1
Entrée 1
Dessert 2
Each time a decision is made the
Dessert 3
outcome is multiplied by a factor Salad 2 equal to the number of choices. Dessert 1
(Order is not important)
Entrée 2
Dessert 2 Dessert 3
Dessert 1
Entrée 1
Dessert 2
Dessert 3 Salad 3 Dessert 1
Entrée 2
Dessert 2 Dessert 3 Rule: The total number of possible outcomes of a series of decisions, making selections from various categories is found by multiplying the number of choices for each decision. Example 1: Find the possible serial numbers formed with the following restrictions: The first two spaces must be a consonant, the next three are non-zero numbers, the last space is a vowel. a) No repeats of letters or numbers. b) Repeats OK. a) 21 X 20 X 9 X 8 X 7 X 5 = 1,058,400 b) 21 X 21 X 9 X 9 X 9 X 5 = 1,607,445 Factorials: Definiton: n! = n(n-1)(n-2)(n-3) ... (3)(2)(1) There is a group of 4 people what want 4 different officer positions in a club. The positions are: President, Vice president, Secretary and Treasurer. How many different combinations can you have of officers? 4 X 3 X 2 X 1 = 24 The counting method we applied used Factorials: 4! = 24 Definition: 0! = 1 Example 2: Find a)
7! = 5040
b) 10!
= 90
c) (8 – 3 )! = 120 Calculator: 2nd --> Math --> Prob --> !
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