 Let's see if we can derive a formula for combinations. So remember our strategy for counting combinations. First, we count permutations. And then, we determine how many permutations correspond to a single combination. So let's think about this. Suppose we select n objects from a set of m objects without replacement. And so we find there are m permute n permutations. Now we could have picked the n objects in... Well, let's see. We have to pick one of them as our first choice. A second one is our second. Our third is a third, and so on, down to our nth choice. And so for the first one, we could have picked any one of the n objects. For a second, any of the n minus one, and so on. And so the number of ways we could have picked the n objects is n factorial. And so n factorial times the number of combinations gives us the number of permutations. And so that says the number of combinations will be... And so this gives us a formula for the number of combinations. And it's useful to remember where this comes from. This is because each combination corresponds to n factorial permutations. And so there are m permute n over n factorial combinations. And we'll use the notation m combine n for combinations. Now strictly speaking, we should call this m combine n because we're looking at combinations. But we frequently read this as m choose n. Now let's do a little bit of algebraic simplification. So this m combine n, well that's m permute n over n factorial. And we have a formula for m permute n. That's m factorial divided by m minus n factorial. The whole thing divided by n factorial. And if we do a little bit of simplification, we get... And so we arrive at the result, the number of combinations when n objects are chosen from a set of m objects without replacement is given by the formula. So suppose we're making a committee by selecting five people from a company of 52. How many different committees can we make? So let's do this problem the old way. So to make that committee, we have to choose a first person, a second person, a third, a fourth, and a fifth. And so for each of these we have a certain number of choices from 52 down to 48. And so we have a hack of a lot of choices. Now since the order in which we choose the members of the committee doesn't matter, this is actually a combination. And so each combination we formed by selecting one of the people first, another one second, and so on. And so each combination could have been selected in 120 different ways. And so 120 times the number of combinations gives us the number of permutations. And so the number of combinations will be still a whole bunch. But what if we use our formula? Remember we can't use our formula until we determine that we're doing our selection without replacement. That's an additional verification step we have to make. And so since you can't pick the same person twice, this selection is done without replacement and our formula is relevant. And so the number of combinations when n objects is chosen from a set of m objects without replacement is given by the formula. And so we have 52 people we're choosing from and we're going to select 5 of them. So we want 52 combined 5. That's 52 factorial divided by 5 factorial times 52 minus 5 factorial. That's 52 factorial divided by 5 factorial times 47 factorial. And this 52 factorial, well that's the product of the numbers from 1 through 52. That's quite a big number. 5 factorial, well that's the product of the numbers from 1 through 5. That's not too bad. And then 47 factorial will be. And when we simplify this horrifying mess we get our answer. And this example illustrates a very important point. Never compute factorials unless you have to. And even then you should hesitate. So how can we avoid computing factorials when they show up in our formulas? Well let's take a look at that. So again we're still trying to find 52 factorial divided by 5 factorial times 47 factorial. But 52 factorial, well that's really the product of the numbers from 1 through 52 or from 52 down to 1. 5 factorial is the product of the numbers from 1 through 5. And 47 factorial goes from 1 up to 47. And the thing to recognize here is that these tail factors from 47 down to 1 are the same in numerator and denominator. And so if we remove those common factors then what we have left is and these are much easier to calculate. And it's worth comparing this computation to the old way. In the old way we calculated this product 52 times 51 times 50 times 49 times 48. Well that's our numerator. And we calculated our denominator which turned out to be the product of the numbers from 1 through 5. And then we divided the one by the other. So again even though we have a formula for calculating the number of combinations and even though this problem was specifically written so that we could use the formula we didn't actually need it. The way that we had been doing will work just as effectively.