Combinations and Permutations

What's the Difference? If the order doesn't matter, it is a Combination If the order does matter it is a Permutation Combinations: Drawing or combination of quantity without order of objects or things. Example: "My fruit cocktail is a combination of mangoes, papayas and bananas". I don't really care what order the fruits are in, they could also be "bananas, papayas and mangoes" or "papayas, mangoes and bananas" etc., its the same fruit cocktail. Permutations: Drawing or combination of quantity with order of objects or things. Example: "The combination to the emergency number is 911". Now I do care about the order. "191" won't work, nor will "119". It has to be exactly 9 -1 -1. Notice: in Mathematics: If the order doesn't matter, it is a Combination. If the order does matter it is a Permutation. Remember Permutation = Position! There are also two types of combinations (remember the order does not matter now): Repetition is Allowed: such as coins in your pocket (4, 4, 4.10, 10 etc.) No Repetition: such as lottery numbers (2.14,15.27,30.33) Combinations with repetition coming soon ... Combinations without Repetition That's how lotteries work. The numbers are drawn one at a time, and if you have the lucky numbers (no matter what order) you win! Formula and different Notations c(n, r)  =  ncr  =  ncr  =  ( n r )  =  n! r!(n − r)! with: c the combination n the total number of items and r the number taken/chosen at one time (once) Here some examples: The oder doesn't matter!!! We abandon the order of objects. We draw three items (numbered 1, 2, 3) in the bag of 10 different objects. How many opportunities, regardless the arrival sequence of objects do we have? The answer: 10c3  =  10! 3!(10 − 3)!  =  10 · 9 · 8 · 7! 3! · 7!  =  120 Choosing 3 balls out of 16, or choosing 13 balls out of 16 have the same number of combinations. The answer: 16c3  =  16! 3!(16 − 3)!  =  16 · 15 · 14 · 13! 3! · 13!  =  560 16c13  =  16! 13!(16 − 13)!  =  16 · 15 · 14 · 13! 13! · 3!  =  560 Top | Permutations | Go to Exercises