Hey, everyone. When finding the number of permutations of 2 letters out of these three letters, knowing that the order matters, whenever I list these out, I end up with 6 permutations. But when considering the combinations of 2 letters out of these three letters, knowing that the order does not matter, I only get 3 combinations. Now just seeing that these give me different numbers, you might realize that we're going to have to calculate these two things differently. But don't worry. We're not just going to have to learn a brand new equation for combinations because we're just going to take our permutations formula and alter it slightly in order to account for the fact that order doesn't matter in order to get our new combinations formula. So with that in mind, let's go ahead and get started.

Now, when working with our combinations, if we just plug in our values for n and r, we end up getting 3!1!, which is just equal to 6. Now for our combinations formula, it looks almost identical to our permutations formula, but now we're just dividing by our factorial. So you'll see that I have my permutations formula right here, and I'm just going to add this extra factor of r factorial on the bottom here. So whenever I compute my combinations formula, plugging in my values for n and r, I end up getting 3!1!•2!, which gives me my answer of 3. Now with this formula, remember that n is always going to be your largest number because this represents your total and r is the number of things you're choosing out of that total.

Now something else that might seem familiar here is this combinations looks similar to our permutations, with the c with the n and the r as subscripts or c with n and r in parentheses. Both of these can be read as the number of combinations of r things taken n at a time.

Now that we know what our combinations formula is, let's get a bit more practice using it by working through some examples together. So looking at this first example, we're told that an ice cream shop has 32 flavors and we need to pick 2 flavors to blend into a milkshake. We wanna know how many possible ways we could do this. Now remember, this is a combinations problem because the order doesn't matter. A chocolate and vanilla milkshake is the same thing as a vanilla and chocolate milkshake.

Now the first thing we want to do here is identify our values for n and r. Remember that n is always going to be the largest number, our total, so here my value for n is 32. Then this 2 right here represents my value for r. So plugging this into my combinations notation, I get 32 c₂, and then I'm just plugging these numbers into the formula. So I start with 32 factorial on the top, your largest number, and then I have 32 minus 2 factorial. And don't forget that extra factor of r factorial on the bottom there.

Now, simplifying this a bit more, I have 32 factorial on the top divided by 30 factorial times 2 factorial. Now here, since we have multiple factorials in that denominator there, remember, we can rewrite our numerator in order to cancel the highest factorial in the denominator. So here, we wanna cancel this 30 factorial, so let's rewrite that numerator to cancel it. So 32 factorial can be rewritten as 32 times 31 times 30 factorial. And then in my denominator, I have 30 factorial times 2 factorial. Now those 30 factorials will cancel, leaving me on the top with 32 times 31. And then on the bottom, expanding that 2 factorial, that's just 2 times 1. Now, actually multiplying that out gives me 992 on the top divided by 2 on the bottom, giving me a final answer of 496 different possible ways we can select our flavors here.

Let's move on to our next example here. Here we're asked how many different teams of 4 people can be formed from a group of 9 people. So let's first identify our values of n and r. Now remember n is your highest number, so this 9 right here, and then r is our other number, so 4. Now writing this as a combination, we have 9 c₄, and we can plug in those values to our formula. So starting with n on the top, of course, we have 9 factorial divided by 9 minus 4 factorial times 4 factorial. Now simplifying this a bit further, we end up with 9 factorial over 5 factorial times 4 factorial.

Now, again, we want to cancel the highest factorial in the denominator here by rewriting that numerator. So here, I'm going to rewrite my numerator as 9 times 8 times 7 times 6 times 5 factorial. And then on the bottom, I have that 5 factorial and 4 factorial, but those 5 factorials are going to cancel. So on the top, I am left with 9 times 8 times 7 times 6. And on the bottom, I have that 4 factorial, which expanded out is 4 times 3 times 2 times 1. Now multiplying that out, I get a rather high number in my numerator here, 3,024, and then on the bottom, I get 24. Now actually dividing that gives me a final answer of 126 different teams of 4 people that can be formed from that 9. Now that we know how to calculate combinations, let's get some more practice. Thanks for watching and let me know if you have questions.