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                                                 8!   Calculator: 2nd   -->   Math   -->   Prob   -->   !   Back to Counting and Probability Main Page Back to the Survey of Math Ideas Home Page e-mail Questions and Suggestions