In this video we discuss what are permutations and combinations, how to use them and the differences between permutations and combinations is explained. Transcript/notes
Permutations and combinations
In order to know all of the possible outcomes for a sequence of events, permutations or combinations are often used.
A permutation is an arrangement of a given number of objects in a specific order, and combinations are used when the order is not important. Here are the main formulas for both of these, for permutations we have big P, n r equals n factorial divided by the quantity n minus r factorial. And for combinations we have big C, n r equals n factorial divided by the quantity n minus r factorial times r factorial.
As a quick review, factorials are basically multiplication countdowns, for instance, 5 factorial equals 5 times 4 times 3 times 2 times 1, and 11 factorial is written on the screen and a zero factorial is equal to one.
Back to our formulas, the n in both of these refers to the total number of objects or options and the r refers to the number of those objects or options being used in the question, and r is less than or equal to n.
As an example, let’s say you are in charge of a concert and you have 8 artists to choose from, you need to select 4 of them to perform, and the order of appearance is important. How many different possibilities are there?
We are going to use the permutation formula, because order of appearance is important. Our n value is 8, because we have 8 different options or artists to choose from. And our r value is 4, because that is how many of the options or artists we need to perform. So, our equation is P, 8, 4, equals 8 factorial divided by 8 minus 4 factorial, which after cancelling out calculates out to 8 times 7 times 6 times 5, which equals 1680 different possibilities.
Now let’s take the same example, 8 artists to choose from and you are selecting 4 of them to perform, but the order of appearance is not important. So, in this situation, we will use the combination formula. So, again, our n value is 8, the total number of options, and r value is 4, the number of options being used. Our equation is C, 8, 4 equals 8 factorial divided by 8 minus 4 factorial times 4 factorial. Going through and doing our math we get 70.
The only difference between these formulas is the r factorial in the denominator here in the combination formula. And in combinations, since order or arrangement is not important, the r factorial divides out the duplicates. For instance, if the order of the 4 performers were listed here, and we just reversed the order here, or, mixed up the order here, in permutations, these are different, because the order is important, but in combinations, these are the same, because the order is not important.
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