By Titu Andreescu

This special approach to combinatorics is established round unconventional, essay-type combinatorial examples, by way of a few conscientiously chosen, hard difficulties and large discussions in their recommendations. Topics encompass variations and mixtures, binomial coefficients and their purposes, bijections, inclusions and exclusions, and producing functions. every one bankruptcy good points fully-worked problems, including many from Olympiads and different competitions, besides as a variety of problems original to the authors; at the end of every bankruptcy are extra exercises to strengthen understanding, encourage creativity, and build a repertory of problem-solving techniques. The authors' past textual content, "102 Combinatorial Problems," makes a good better half quantity to the current paintings, which is ideal for Olympiad members and coaches, complex highschool scholars, undergraduates, and faculty instructors. The book's strange difficulties and examples will interest pro mathematicians to boot. "A route to Combinatorics for Undergraduates" is a full of life creation not just to combinatorics, yet to mathematical ingenuity, rigor, and the enjoyment of fixing puzzles.

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**Example text**

Proof. Let P and Q be the permutation matrices corresponding to the permutations σ = k1 k2 · · · kn and π = l1 l2 · · · ln , respectively. Then, in G(P ∗ Q), there are exactly n paths of length 2 from black vertices to white vertices, each of weight equal to 1 · 1 = 1, and these paths have no vertices in common. In G(P Q), there is exactly one edge from each black vertex to each white vertex, and these edges all have weight equal to 1. More precisely, for each i = 1, 2, . . , n, there is an edge of weight 1 from black vertex i to white vertex lki .

Cm , not all of which are zero, such that c1 u(1) + c2 u(2) + · · · + cm u(m) = 0, that is, the zero vector can be written as a nontrivial linear combination of u(1) , u(2) , . . , u(m) . For example, the vectors (1, 4), (3, −1), and (3, 5) in ℜ2 are linearly dependent since 3(1, 4) + 1(3, −2) − 2(3, 5) = (0, 0). Vectors are linearly independent provided they are not linearly dependent. The vectors u(1) , u(2) , . . , u(m) are a basis of V provided they are linearly independent and span V .

3. Prove that there does not exist a regular graph of degree k with n vertices if both n and k are odd. 4. Determine the chromatic numbers of the following graphs: (a) the graph obtained from Kn by removing an edge; (b) the graph obtained from Kn by removing two edges (there are 24 CHAPTER 1. INTRODUCTION two possibilities: the removed edges may or may not have a vertex in common); (c) the graph obtained from a tree by adding a new edge (the new edge may create either a cycle of even length or a cycle of odd length).