And this problem, we're trying to figure out the probability that Xerxes arrives first and Regina arrives last. Now, the first thing to note is that there are nine people. So if we list off nine different spaces, there's nine spaces and now the order in which they arrive could be any order. So for the first spot there are nine different ways that someone can show up, Anyone can show up first and then once someone has shown up first, the person who arrives second, there are eight different ways to choose that person. Similarly, the person who arrives third, there are seven people remaining, so there's seven ways to choose that and so on. And so there are actually nine factorial ways that the people can arrive to the party. Now if xerxes needs to be in the first spot and Regina needs to be in the last spot than in these remaining seven spaces, we can put any people, so there can be any ordering between xerxes and Regina. So there is seven factorial ways to order the people between xerxes and Regina. So the probability that we end up with is seven factorial divided by nine factorial. So that is seven factorial. And remember that nine factorial can be written as nine times eight times seven factorial. The seven factorial are going to cancel. We get 1/7 times eight which is equal 1/72 which is equal to approximately zero point 014 and that's it
