The following is a brief overview of permutations. As a general definition, a permutation is “usually understood to be a sequence containing each element from a finite set once, and only once.”[1] A very basic example of a permutation would be as follows:
Find the number of ways in which you can arrange 6 numbers from the set of numbers {1,2,3,4,5,6,7,8,9,10}.
For starters, we would denote the answer to this question as 10_P_6. That is to say, we are selecting permutations that include 6 out of 10 numbers from the set, without repeating any of the numbers. It is possible to find the number of permutations by writing out various sequences of numbers manually. However, this is extremely time-consuming (especially for a set as large as 10 items, like the one we’re using). Alternatively, we can develop a formula to find out the number of possible permutations. To do so, we need to know the following.
First, what are the total number of permutations REGARDLESS of the limit we are given (6 out of 10 numbers from the set)? To find this number, we simply calculate the factorial of the number of items in the given set. In our case, we need to find the factorial of 10, which is written as 10!.
10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800
Obviously this is a lot of possible sequences, and you should be able to see how creating a formula will save you time in calculating the final answer. The next step is to narrow down our answer to include our limit (6 out of 10 numbers from the set). Since we are only looking for sequences of 6 numbers, we need to divide the total number of permutations (10!) by the number of items in each sequence we will include in the final answer (10 – 6 = 4!). The whole process will look like this:
10_P_6 = 10! / (10-6)!
= 10! / 4!
= (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / (4 x 3 x 2 x 1)
Essentially, this is the same as writing (10 x 9 x 8 x 7 x 6 x 5), which equals 151,200. So, the final answer would be:
10_P_6 = 151,200
Which suggests that there are 151,200 possible permutations when selecting 6 out of 10 numbers from the set {1,2,3,4,5,6,7,8,9,10}. Permutations can be used to select random data from large sets, without going through the hassle of writing all the possible sequences out by hand. As a real-world example, permutations are often times used in statistics to find the number of defects in a manufacturing process. For more information on permutations, as well as combinations, I would suggest visiting mathforum.org.[2]
References:
- “Permutation.” Wikipedia. 11 February 2009. URL http://en.wikipedia.org/wiki/Permutation
- “Math Forum: Ask Doctor Math FAQ: Permutations and Combinations.” Mathforum.org. 11 February 2009. URL http://mathforum.org/dr.math/faq/faq.comb.perm.html#note
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