[SOUND] Hello, and welcome to our course which is entitled Introduction to Enumerative Combinatorics. My name is and I'm an Associate Professor of the National Research University Higher School of Economics Department of Mathematics in Moscow, Russia. The very first thing that you learn about mathematics when you're a kid is that math is about counting things. That mathematicians are counting something. Well, I think that maybe they're doing computations with very long numbers all day long. Well, that's what you think about math and about mathematicians when you were a child. But later, you understand, as you study more and more mathematics, that this is not completely true. That much of mathematics is about proving something. Much of math is about constructing new theories and finding out properties of certain mathematical objects, and so on, and so on. But there is one branch of mathematics, namely combinatorics, for which is it essentially true that combinatorics is about describing cardinalities of finances, and well, so it is about counting certain objects, having certain given properties. So a typical question in combinatorics starts with the words, in how many ways can we do something. Well, let me give a couple of examples of typical commentorial questions. So first of them is, in how many ways can we pick six numbers. Six different numbers out of let's say the first 50? Can we pick sic distinct numbers. In the range. From 1 to 50. A very practical question for everyone who had ever played a lottery. So many lotteries are about getting the six random numbers in this range. So if you want to compute the odds of winning, then winning a jackpot, you need to find this number of different ways of getting such a topple of six numbers. Well, so in fact, many combinatorial questions were historically motivated by gambling, by some games of chance, such as card games, or games of dice, and so on. So probably, you have already dealt with some combinatorial questions when you were starting probability theory. Well, another question is, in how many ways can you split say a $20 bill into smaller bills of $1, $5 and $10? Or another question is, suppose you're climbing a ladder and here's your ladder. And you can climb either one step or two steps at once. So a typical way of climbing a ladder would be. And here, climb one step up, two steps up, and then another two steps, one step, and one step, and here you are on the top. So if you are proceeding in this way, in how many ways can you climb this ladder? If we're going either one or two steps up at once. Of course, we know how many steps does the ladder have. Okay, so we'll be dealing with some questions of finding cardinalities of finances, finding the number of ways of doing something. Before we start, let me tell you a couple of words about this course. This course consists of eight lectures. And the first half of our course, lectures from one to four will be devoted to basic principles of enumeration. And we'll be dealing with the basic commentorial problems involving permutations, binomial coefficients in their many occurences and sequences defined by layers of relations such as the Fibonacci sequence. And some more involved in current relations, such as catalogue numbers. And in the second half of our course we'll be dealing with a very powerful method of solving combinatorial problems, namely with generating functions and their uses in various combinatorial problems. And in particular, we'll speak a lot about partitions, and about variantions of binomial coefficients, namely the Gaussian binomial coefficients, or Q binomial coefficients as they are sometimes called. So I hope you will enjoy our course and let's talk. [SOUND]