Applied Statistical and Optimization Models Topic 06-02: Permutations and Combinations

Objectives Difference between permutations and combinations Factorials Calculating the odds of winning the lottery

Probability Permutations – “Number of ‘Combinations’ when the order matters” With repetitions allowed With repetitions NOT allowed Answer: Permutation P= 40∙39∙38=59,280 Permutations refer to the number of “combinations” where the order matters when repetitions are not allowed. General formula for permutations w/o repetitions: which reads as “the number of permutations of size r from the population of n is equal to n factorial over (n-r) factorial.” Note: 0!=1 How many 3-digit (r=3) permutations are possible in a lock of n=40 numbers (0 – 39)? Answer: Permutation P= nr =403=40∙40∙40=64,000

Probability Combinations – “Number of ‘Combinations’ when the order does not matter” For example, in a lotto game, in which you need to guess right the combination of a certain number from a drawing, the order of the drawing does not matter. The subgroup combinations of size two of the set S={A,B,C} allows for only three combinations, which are AB, AC, BC (Combinations) For comparison purposes, there are six permutations, which are AB, AC, BA, BC, CA, CB (Permutations) The formula for the number of subgroup combinations of size r obtainable from a population of n objects is

Probability Permutations: You have five Lego blocks, each in a different color. How many three-block tower permutations can you build? This assumes that different color sequences stand for different towers. Answer: Combinations: You have five Lego blocks, each in a different color. How many three-block tower combinations can you build? This assumes that each tower’s color sequence does not matter.

Probability Explaining the Odds of Winning the Georgia Lottery There are 56C5 combinations to get 5 out of 56 right (remember, the order does not matter). There are 3,819,816 combinations of drawing five out of 56. The probability to get 5 out of 56 right is therefore 1/ 3,819,816. There is additionally (multiplication rule!) a 45/46 chance to miss the super ball. The total number of combinations to win “5 out of 56” and “not winning the superball” is therefore In the Georgia Lottery, you can play a game called “Mega Millions” where you can win $250,000 if you guess right five out of 56 numbers, but miss the superball drawn from 46. Likewise, the number of combinations to 5 out of 56 and the superball is

Probability Explaining the Odds of Winning the Georgia Lottery (Contd.) Likewise, the total number of combinations to get “5 out of 56” and “1 out of 46” is