Hey, everyone. Up to this point, we've been working with the permutations of different objects. So when considering the permutations of two letters from a, b, and c, I would consider a b, but I would also consider b a. These are 2 different permutations because they're in 2 different orders. But now we're going to look at a new way to organize objects called a combination. Distinguishing between permutations and combinations can be tricky, especially because these words are sometimes used interchangeably in everyday language. But here, I'm going to walk you through exactly what the mathematical difference is between these two things and how to distinguish between them by considering just one thing, whether or not the order of these objects matters. So let's go ahead and jump right in here.
Now, in working with permutations, we saw that they were a different way to arrange objects in a particular order. So we know that when working with permutations, the order definitely does matter. So when writing all the permutations of two letters from a, b, and c, I would want to consider b, c, but also c, b, and a c, but also c a. Because these are in completely different orders, they represent different permutations. But when working with combinations, combinations are simply a way to group objects. So if I consider groups of 2 letters from a, b, and c, I might consider ab, bc, ac, but nothing else because when working with ab, this is the same group of letters as if I were to have b a. So when working with combinations the order does not matter at all.
This might seem a little abstract, so let's go ahead and look at some different examples and determine whether we're working with a permutation or a combination. So looking at this first example here, I'm told that an ice cream shop has 32 different flavors, and we need to pick 2 flavors to blend into a milkshake. Now we want to know how many possible ways we can select these flavors. We want to ask ourselves one question: Does the order of my objects matter? Well, if I'm blending up a milkshake, I might want to consider the outcome. So if I pick chocolate and vanilla and I blend that up, wouldn't that be the same milkshake as if I picked vanilla and then chocolate? Well, it would. So my outcome would be no different even if I changed the order. So since the order does not affect the outcome, here I'm simply working with a combination. The order does not matter.
Let's look at our next example. Here we're asked how many ways could a photographer line up the members of a family of 5? Well, if I were taking this photo, I might consider a couple of different things. I might consider the height of the people in it or the outfits that they're wearing. So if I put one person on the end in one of those photos, but then I move them to the other side in another photo, this would be a completely different photo. So if I change the order, that affects the outcome. So because the order affects the outcome, that tells me that order does matter, and here we're working with a permutation.
Let's look at one final example here. Here we're asked how many different teams of 4 people can be formed from a group of 9 people. Well, let's say that I pick 4 people out of this group of 9. If I pick this person first instead, is that going to change what my team is? Well, it's not going to change it at all besides maybe starting a fight between teammates. So because the order does not have an impact on the outcome, here the order does not matter, and we're working with a combination.
So now that we know how to distinguish between permutations and combinations, let's keep going. Thanks for watching, and I'll see you in the next one.