Overview:
Permutations and factorials are closely related mathematical concepts. Permutations relate to the order of objects, while factorials involve all possible outcomes of an event. If items are ordered in a particular way, factorials determine the number of times they can be ordered.
What Are Permutations?
Permutations are rankings where the order matters. Suppose a music producer is choosing the order of songs on a CD released from a new artist. Only one song can be in each track. If there are 10 songs that have been recorded, there are 10 possibilities for the opening track. Since one song has already been chosen, there are 9 left for the second track. For the third track, there are 8 left, and so on, until the last song remains. The way to calculate the number of permutations is 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 3, 628, 800.
What Are Factorials?
Factorials involve all possible outcomes of an event. They are multiplications that start with the number of events multiplied by the number of events -1 and so on, until there is only 1 left. In the example where there were 10 songs on a side of the CD, 10! (read 10 factorial) is 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1. The number 0! is equal to 1, much in the same way that x0 is equal to 1. It is a mathematical convention that makes calculations easier.
What Are Combinations?
Combinations are rankings where the order doesn’t matter. For example, suppose 5 people are chosen out of a group of 10 to be on a committee. In this case, it doesn’t matter in what order they are chosen, because the end result is the same. An entire group of people are chosen. If the order mattered, it would be 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 = 30240. Since the order does matter, it will be a subset of the 30240. How many of the groups of 5 contain the same members in whatever order? That can be answered by the factorial 5!, which is 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 120. 30240/120 = 252.
How Are They Different?
Suppose there are 11 runners in a race. Most of the time, all 11 runners aren’t given places, just the first 3. There are 11 possibilities for first place, 10 possibilities for second place, and 9 possibilities for third place, or 990 permutations in the first 3 places. If all runners were given places, that would lead to 11! places, or 39,916,800 possible combinations.
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