If you have one clean shirt and one clean pair of pants, then you have a super easy decision when getting ready in the morning. But what if you just did laundry and you actually have 3 clean shirts and 4 clean pairs of pants? How many different possible outfits could you come up with then? Well, this is actually the exact sort of thing that you'll be asked to do: find the number of total possible outcomes when faced with multiple options for multiple different things. This might sound like it could be overwhelming or time-consuming having to come up with all of these, so let's go ahead and just jump right in here. Now with our 3 shirts and 4 pairs of pants, if I take that first color shirt and pair it with all four different colors of my pants and then do the same thing for my 2nd and 3rd shirts, I see that I have 12 total possible outfits that I could make on that day. But I can actually come to this conclusion much easier using the fundamental counting principle. The fundamental counting principle tells us that if there are m possible choices for one thing and n possible choices for another thing, I can simply multiply these together, m⋅n, to get the number of total possible choices for both of those things together. So, for my 3 shirts and 4 pairs of pants here, I can simply take 3 and multiply it by 4 to get my total of 12 possible outfits without having to actually come up with them and count them out.

So now that we've seen the fundamental counting principle, let's go ahead and work through some more examples together. Looking at this first example here, I see that a menu lists 4 appetizers and 6 entrees. And I want to know how many different meals with both an appetizer and an entree do we have to choose from. So, my first choice to make is of those 4 appetizers, so I have 4 possible choices there. Then for my second thing, I have 6 entrees to choose from, so I have 6 possible choices there. Now if I multiply those 2 together, 4 times 6, I get my number of total possible choices of a meal with an appetizer and an entree, and we're done here.

Now let's look at our second example. Here, we're asked to find how many possible outcomes there are if we flip a coin and then roll a 6-sided die. Well, something that you might notice here is that we're not really making a choice. If I flip a coin, there are 2 things that could happen: I could get heads or tails, but I'm not really choosing here. And that's okay because the fundamental counting principle also applies to outcomes of events. So if there are m possible outcomes of one event and n possible outcomes of another event, I can still multiply m⋅n to get the number of total possible outcomes of both events. So let's go ahead and do that here. When flipping a coin, we know that we can get heads or tails. There are only 2 possible outcomes here, so I'm going to take that 2 and then multiply it by the number of total possible outcomes of that second event. When I roll a 6-sided die, I could get any number 1 through 6, so I know that I have 6 total possible outcomes here. So multiplying those 2 together, 2 times 6, I get my total of 12 possible outcomes of both of these events together.

Let's take a look at one final example here. Here, we're asked how many different outfits can be made from 4 shirts, 5 pairs of pants, and 3 pairs of shoes. So the first choice I have to make is what shirt I'm going to wear out of these 4, so I know that I have 4 possible choices there. Then for my second thing, I have 5 pairs of pants, so I want to multiply 4 times 5. But since I have a third thing here, what exactly do I need to do? Well, the fundamental counting principle doesn't just apply to 2 events. It can also apply to any number of events. So if I have more than 2 things that I'm choosing from, I'm just going to continue to multiply by the number of options of each thing, and that will give us our total. So here, since I have 3 pairs of shoes, I'm going to continue to multiply, and I'm going to multiply 4 times 5 times 3, which gives me 60, my number of total possible options of those three things together. Now that we've seen the fundamental counting principle and know how to use it, let's get some more practice. Thanks for watching, and I'll see you in the next one.