By Titu Andreescu

This special approach to combinatorics is based round unconventional, essay-type combinatorial examples, via a couple of conscientiously chosen, difficult difficulties and huge discussions in their options. Topics encompass diversifications and mixtures, binomial coefficients and their functions, bijections, inclusions and exclusions, and producing functions. each one bankruptcy beneficial properties fully-worked problems, including many from Olympiads and different competitions, in addition as a variety of problems original to the authors; at the end of every bankruptcy are additional exercises to strengthen understanding, encourage creativity, and build a repertory of problem-solving techniques. The authors' past textual content, "102 Combinatorial Problems," makes an outstanding significant other quantity to the current paintings, which is ideal for Olympiad contributors and coaches, complex highschool scholars, undergraduates, and faculty instructors. The book's strange difficulties and examples will interest professional 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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First Solution: It is perhaps easier to think in terms of n knights, where n > 4. We first count the ways of selecting three knights if there are no restrictions. We can use the multiplication principle. We have n ways to choose the first knight, then n- l ways to choose the second knight, then n - 2 ways to choose the third knight. But the order of choosing these three knights to form the triplet does not matter. A triplet A, B, C can be chosen in 6 ways, namely, ABC, ACB, BAC, BCA, CAB, CBA. Therefore, each triplet is repeated six times in n(n - 1J(n -2) ways to the selection procedure.

Hence the answer is 21�O + � = 26. 14. Let n be an integer with n > 3. 9). Three points Pi, Pi , and Pic are randomly chosen, where i, j, and k are • • • Counting Strategies 38 1 and n, inclusive. distinct integers between that the triangle PiPj 1'k What is the probability is obtuse? Because w is the circumcircle of the triangle Solution: PiPj Pk, the center of w is outside the triangle if and only if the triangle is obtuse. We consider two cases. + 1 is even, say n = 2m m > 2, for some positive integer is diametrically opposite to then PI.

Claudia has cans of paint in eight different colors. She wants to paint the four unit squares of a 2 x 2 board in such a way that neighboring unit squares are painted in different colors. 20 Counting Strategies Determine the number of distinct coloring schemes Claudia can make. Two coloring schemes are considered the same if one can be obtained from the other by rotation. Solution: Claudia needs at least two and at most four colors. 7. 7. D. 8. In other words, each coloring scheme in this case is counted four times, considering rotations.