First, we show some examples that go from purely combinatorial problems to geometric ones, and how invariants are used to solve each of them.
Then, we present in depth how coloring techniques are used in problems related to boards, especially those concerning some type of tiling. Then, we introduce some concepts related to game theory, and show how some problems of this kind can be solved using invariants. At the end of the chapter, 23 problems are given for the reader to practice. This chapter contains an introduction to graph theory.
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The purpose of this chapter is not to give a purely theoretical introduction, so most concepts are shown with problems whose solution involves this idea. First, basic concepts are presented, and important properties of graphs are proven. However, we show how international olympiad problems can be solved using only these basic tools. Then, we study the concepts of connectedness in graphs and properties of trees.
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We introduce bipartite graphs and some of their properties that can be used frequently for problem-solving particularly for problems involving boards. At the end of the chapter, 18 problems are shown. This chapter presents how to use functions in combinatorial problems.
The main objective is that the reader sees how constructing functions and observe basic properties of them can help to deduce meaningful information about the problem. This is done in the first section. Then, we study permutations. We analyze their structure and some of their properties that make working with them easier. Once this machinery is set up, we show examples where this can be helpful.
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In the next section, we introduce the concept of counting twice. This technique has been used throughout the book, but now we view this properly from a functional point of view, with examples from math olympiads of increasing difficulty. This shows the power that this innocent-looking technique has. At the end of the chapter, 19 problems are presented.
This chapter contains an introduction to generating functions. We present generating functions as a way to deal with recursive equations that appear in combinatorial problems. First, we define these functions and their basic properties, and give examples of many basic generating functions. Then, we show how they can be used to obtain a closed formula to the Fibonacci numbers, and how this technique extends to similar sequences. As a more complicated example, we show how to obtain a closed formula for the Catalan numbers.
This is done both via generating functions and with a purely combinatorial approach. Then, we introduce the concept of the derivative, and how it can make manipulations of generating functions much easier. Namely, we explain when it is possible to evaluate them at a point and when an actual function can be represented as a generating function.
At the end of the chapter, we present 18 problems for the reader. This chapter studies partitions, and shows how the techniques presented throughout the book can be applied for this purpose. First we study partitions of integers, and their Ferrer diagrams. Then, we present the Stirling numbers of the first kind by their algebraic properties. Central state college has three sections of Math Could there be any way to visualize it in such a way?
Also, is I came across this question on a GRE practice exam.
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If there are 20 people with 17 men and 3 women and they seat themselves randomly at 4 Tables A,B,C,D with 5 people each. With all arrangements being equally likely, what is the probability that If a computer prints these arrangements at the rate of per second, how long will this take?
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