All right. Then let's finish up with a few exercises that you might try to check your understanding of the material in this lecture. so the first one is an exercise that has to do with calculating leaves in Cayley Trees which you recall are unordered rooted trees and shows it's, it's a node which gives much of the results and you have to fill in some of the details. but that's a good exercise to check the use of bivariate generated functions and symbolic methods for a useful problem. the second one is recreational mathematics. so if you take all the numbers that are formed the decimal numbers, integers, and digit 1 is used once, digit 2 is used twice, and so forth, digit 9 used nine times. so that's 45-digit numbers. And you take all of them and add them together. if you use if you do that with a extended precision arithmetic package you'll find out that you have a huge number of nines in a row, and actually lots of other nines. and so you're supposed to explain that. and and so this was actually discovered in 1150 AD and so maybe with ana-, analytic combinatorics we can better understand it. and so, read again the chapter in the text about multivariate generating functions. And again, there's quite a bit of advanced material in there. It's not attended that everybody read every page but rather to get a good idea of what's in there and read more detail about what you find interesting. write up solutions to those 2 exercises and try the programming exercise. Again, just to validate the math. generate some random permutations and check that the number of cycles that you get in those permutations is, is about HN. even, even for Math it's worthwhile to validate the Math that we do to better understand the der-, derivations. so those are the exercises for or combinatorial parameters lecture.