So, here's a few exercises that you could use to test your understanding of material in this section. So, first one is just a calculation. So, how many people would you think, if you were a paranoid professor and you had a big class, how big would the class be to be 99% sure that [INAUDIBLE] If you ask everybody in the class, you're going to find two that have the same birthday. So that's just a fun calculation. So this next one is kind of a co-monitorial thing, it's actually the basis for analysis of linear probing And it ties together some of the calculations that we did in this section having to do with caley trees. And it's actually easier than it looks, so it's proving this generalization of the binomial theorem due to able. And it's worthwhile doing this calculation. And then here's an exercise that isn't in the book that, but it should be there, so I numbered it number 99, and it's to show that the probability if you take a random mapping of size N, what's the probability that it doesn't have any singleton cycles, nothing that maps to itself. Amazingly, it turns out to be n over e. The same as the derangement problem for permutations. There's a no good reason that these things should be the same but it turns out not to be the same. So that's a nice exercise for you to take a look at. So to finish up, read that part of the text. Good idea to run empirical tests to see if the analysis works, and you'll find that it does. And it's also a good idea to take a look at properties of mapping and check that the analysis works as well, and then maybe write up the solutions to those exercises. That's the end of Analytic Combinatorics part one. We've done a pretty full survey of basic techniques, introduced analytic combinatorics. And showing how it applies to the study of basic combinatorial structures, like trees, permutations, strings, words and mappings. We hope that many of you are interested enough in the problems that we've looked at to sign up for analytics combinatorics part two which will start soon. Thanks.