 In general, most probabilities cannot be calculated. Remember, a problem exists whether or not we can solve it. However, in very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very. Very rare cases, we can use permutations and combinations. And so, realistically, you'll have very few opportunities to calculate probabilities this way, except when someone is asking you to calculate a probability that you can calculate this way. What's really important here is not so much the computation of the probabilities but the thought process that leads to it. This leads us to the topic of combinatorial probabilities. A combinatorial probability can be computed when the sample space consists of equally-likely outcomes and the outcomes are permutations. For example, let's say we flip a fair coin ten times. Is the probability of getting exactly five heads in ten flips a combinatorial probability? So, in order to answer this question, we need to consider our sample space. So, if we let our sample space be the sequence of heads and tails, first it appears that all outcomes are equally-likely, and the sequence of heads and tails is a permutation. We can form it by choosing what we have as the first flip, the second flip, and so on. And so this means we can express this as a combinatorial probability. Now, to find the number of ways, we need to ask questions and determine how many possible answers we have. Since we want an event to occur, our question should be about how it will occur. So, let's ask the combinatorial question, how many ways can a coin land heads exactly five times in ten flips? So, suppose the coin did actually land heads exactly five times in ten flips. So, the combinatorial question is how did we get that? And here's the most difficult part of combinatorial probability, and really the most difficult part of anything is asking the right questions. The question we might ask is when did the first heads occur, when did the second, and so on. And so, let's set up our sequence of questions, first head, second head, third head, fourth, and fifth. And so, the first time the coin landed heads could be any one of the ten flips. Wait a minute, no, no, it can't be. And if you think about it, the problem is since we have to get five heads, you have to have at least one heads result by the sixth flip, otherwise you can't get five heads in ten flips. So, there's only six possible choices for when that first heads occurs. How about that second heads results? Well, the number of choices for the second time it lands heads depends on when the first flip occurred. Again, since we have to get a total of five heads, if the first time we got heads was on the sixth flip, then all the results after that have to be heads. And so, there's only one choice for when we get that second heads. On the other hand, if the first time we got a heads result was on the first flip, then the second time we got a heads result could have been on the second through seventh flips. It can't occur after the seventh flip, because then we wouldn't be able to get five heads. And so, here we have six possibilities for when the second heads result was obtained. And so, the problem is that the number of choices for the second time it lands heads depends on when that first flip occurred, and this is really too complicated for us to analyze. And so, here's a useful bit of insight when looking at these combinatorial probabilities. Don't worry about order at the beginning. So, instead of asking when did the coin land heads the first time, we'll ask a less ordered question, when was one of the times the coin landed heads, when was another time, when was the third time, and so on. And so now, one of the times the coin landed heads, well, it could have been any one of the ten times, another time the coin landed heads could have been any one of the nine remaining times, and so on. Now again, suppose we have our answers, the coin landed heads on the first, seventh, third, second, and eighth times, and landed tails on all the others. Since we're arranging the answers gives us essentially the same result, we have a combination, and so every combination corresponds to 120 permutations. And so these 30,240 permutations correspond to a lot fewer combinations, and so there will be 252 ways of getting five heads in ten flips. So now let's turn this into a probability. So, our sample space consists of all possible ways the coin could land if we flip it ten times. Let's think about this. We can't switch the outcomes around without getting something different. So for example, result of the third flip can't be the answer to result of the eighth flip. And because of that, we're dealing with permutations. Now since each flip has two possible outcomes, there are two to the tenth permutations. Now, we've calculated already that there's 252 ways of getting exactly five heads in ten flips. There are two to the tenth 1024 possible ways to flip a coin ten times, and so the probability of getting exactly five heads is 252 out of 1024. Now, combinatorial probability is very generic term, and so later on we'll call this type of probability a binomial probability.