Definition of Combination An arrangementof objects in which the orderof selection does NOT matter. Ex: You have to visit three out of your four friends houses: Andrew (A), Betty (B), Carlos (C), Dave (D). What are the different ways to select the 3 houses to visit? Similar to Permutations, there are no repeats. But in contrast to permutations, order does not matter. The keys to Combinations are: A,B,C A,C,D A,B,D B,C,D Each arrangement is one combination of the elements A, B, C, and D. In other words, there are 4 total combinations.
How to Calculate the Total Number of Combinations The total number of ways (without repeats) to choose r objects from a set of n objects (order does NOT matter). Textbook Definition The method from the last slide Ex: Jim had 9 friends and needs to select 4 of them to go on a trip. How many different arrangements are possible?
Combination Example A bag has 7 marbles (blue, green, red, yellow, orange, purple, and black). If you select four marbles at once, how many combinations are possible? How many ways can you arrange 4 marbles from 7? It is clear that selecting BGRY is not different than BRGY. In the 840 permutations, how many times will B, G, R, Y be repeated? How many ways can you arrange B, G, R, Y? If every combination is repeated 24 times, how many are possible? The # of Permutations divided by the factorial of the # of decisions.