 The key problem of the branch of mathematics called combinatorics is how many ways can a set of choices be made? For example, how many different outfits can I select for my wardrobe? How many different routes are there between points A and B? How many different winning hands are there in the game of poker? In order to solve problems like these, imagine answering a sequence of questions. The answers you give to the questions form a permutation. More generally, if you can't switch answers around, or if switching answers around gives you something different, you have a permutation. For example, what do you want for breakfast, lunch, and to wear? And maybe you answer eggs, tacos, and a clown suit. Since switching the answers around gives us something that is different, it's different if you have eggs or tacos for breakfast, and in some cases you can't switch the answer, you certainly couldn't eat a clown suit for lunch, then we have a permutation. Or we could ask a question like, what are the digits of your telephone number? And maybe we have an answer like 555-1212. If we switch these digits around, we get a different telephone number, and so we have a permutation. Who's going to be at the party? Well, maybe it's Ann, Bob, Charlie, and Doug. But if we switch our answers around, say Charlie, Bob, Doug, and Ann, it's still the same group of people, and so this is not a permutation. Or let's take a real world example. A restaurant offers a following meal deal. You can choose nontree, a side, and a drink. Is this a permutation meal or a combination meal? Well chances are, most ordinary people will call this a combination meal. But mathematicians are not ordinary people. So let's think about this. We'll pick an entree, how about fried chicken, a side dish, how about sliced apples, and a drink, mango lassi. Now can we switch our answers around, and the answer is no, we can't switch our answers. Sliced apples will always be a side, never a drink or an entree. And since we can't switch the answers around at all, this is a permutation. So suppose two people are selected to go to a conference. Is this a permutation or a combination? If we pick Agnes and Bob, it's the same as if we pick Bob and Agnes. And so this is a combination. Two people are to be picked, one for a president and the other for a treasurer. Is this a permutation or a combination? So if we pick Agnes for president and Bob for treasurer, it's different than if we pick Agnes for treasurer and Bob for president. So while we can switch our choices around, switching produces a different result, and so this is a permutation. So now we have some way of distinguishing between permutations and combinations. Let's see if we can determine how many we have. And this is all based on what's known as the fundamental counting principle. Suppose we have two choices we can make. If we have m ways we can make the first choice and n ways we can make the second choice, then there are m times n ways to make the two choices. Equivalently, we say there are m times n permutations. So let's go back to that meal permutation. So you can make a meal by choosing one of five entrees, one of 12 sides, and one of seven drinks. How many different meals are possible? So it's helpful to think of writing down our choices as answers. So we have to know what we choose as an entre, as a side, and as a drink. And again, since we can't move our choices, we have a permutation and so we can apply the fundamental counting principle. So for entrees, we have five possibilities. For sides, we have 12 possibilities. And for drinks, we have seven possibilities. And so the fundamental counting principle says that the number of ways we can make a choice of entre, side, and drink is the product 5 times 12 times 7, 420.