By Richard A. Brualdi

This relied on best-seller emphasizes combinatorial ideas–including the pigeon-hole precept, counting strategies, diversifications and combos, Pólya counting, binomial coefficients, inclusion-exclusion precept, producing capabilities and recurrence family members, combinatortial buildings (matchings, designs, graphs), and flows in networks. The 5th version clarifies the exposition all through and provides a wealth of latest routines. Appropriate for one- or two-semester, junior- to senior-level combinatorics courses.

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Extra info for Introductory Combinatorics (5th Edition)

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Ei is even, ao + bo + ... + eo is even. A Nim game that is not balanced is called unbalanced. We say that the ith bit is balanced provided that the sum ai + bi + ... + ei is even, and is unbalanced otherwise. Thus, a balanced game is one in which all bits are balanced, while an unbalanced game is one in which there is at least one unbalanced bit. We then have the following: Player I can win in unbalanced Nim games, and player II can win in balanced Nim games. To see this, we generalize the strategies used in 2-heap Nim.

The requirement that the basket be nonempty, then there would have been 9 or 10 choices for the number of apples depending on whether or not the number of oranges was 0, and we could not have applied the multiplication principle directly. But an alternative solution is the following. Partition the non empty baskets into two parts, Sl and S2, where Sl consists of those baskets with no oranges and S2 consists of those baskets with at least one orange. The size of Sl is 9 (1,2, ... ,9 apples) and the size of S2 by the foregoing reasoning is 6 x 10 = 60.

Show that the following map of 10 countries {I, 2, ... , 1O} can be colored with three but no fewer colors. If the colors used are red, white, and blue, determine the number of different colorings. 22 CHAPTER 1. WHAT IS (OM8INATORI(S? 1 4 7 2 5 8 3 6 9 21. (a) Does there exist a magic hexagon of order 2? That is, is it possible to arrange the numbers 1,2, ... ,7 in the following hexagonal array so that all of the nine "line" sums (the sum of the numbers in the hexagonal boxes penetrated by a line through midpoints of opposite sides) are the same?

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