let's, finally let's look at some exercises that people might want to look

at to check their understanding of the material in this lecture.

the first one is typical of a mathematical model in the real world that

people have studied. This is called the Ehrenfest model.

and it's a situation where you have two urns and balls and at any Then one ball's

chosen at random and switches. and this is a classical model rated

related to a particular physical situation.

And this is an opportunity to look at exponential generating functions for

studying properties of this model. and second one is, a little more detailed

about trigonometric funcitons and how they, relate to, combinatorics.

Again, there's something essential about the kinds of funtions that, generating

functions that, turn up, when we use the symbolic method because the methods in

set and sequences cycle are, are so basic, and it relates to, mathematics,

looked at from other direction, as well. so again read through the chapter on

labeled structures and exponential generating functions, in the text.

Now again, reading this kind of material carefully, is something that might take

months so it's more about, trying to read the descriptive part of the text, in

particular the perspective. and then also to see what kinds of

examples are in there and maybe read some that might peak your interest in more

detail later on. just to know that for future reference

where these problems are treated and how much details required to to treat them.

and then write up solutions to those two exercises.

and then if you're interested, tried this programming exercise and it's very easy

to simulate that Ehrenfest model. and go ahead and plot the distribution

and see if it, if it matches what the theory says, that it should.

this is not quite the same as going out and actually, measuring molecules and

But still it checks, the math checks with the model.

and, it's an interesting exercise. so those are, exercises related to label

structures and exponential generating functions.