Calculating in Two Ways: Fubini’s Principle

  • Titu Andreescu
  • Zuming Feng


So far, most of the problems that we have presented, regardless of the level of difficulty or complexity, could have been solved with direct calculation, that is, each problem could be solved by counting the number of objects (maybe via bijections or recursions). In this Chapter, in order to find x, the number of objects of type A, we will find y, the number of objects of type B, in order to set up equations involving x, after which we will solve for x. In other words, the problems that we analyzed in the earlier chapters are Arithmetic problems, but now we will be doing Algebraic problems by setting up equations and solving them.


Positive Integer Equilateral Triangle Convex Polygon Maximal Chain Isosceles Triangle 
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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Titu Andreescu
    • 1
  • Zuming Feng
    • 2
  1. 1.American Mathematics CompetitionsUniversity of NebraskaLincolnUSA
  2. 2.Department of MathematicsPhillips Exeter AcademyExeterUSA

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