Zuming Feng

Inclusion-Exclusion Principle

The Inclusion-exclusion Principle is helpful for counting the elements of the union of overlapping sets. We consider a few basic examples.

Properties of Binomial Coefficients

[AIME 1987] A given sequence r 1, r 2, ..., r n of distinct real numbers can be put in ascending order by means of one or more “bubble passes.” A bubble pass through a given sequence consists in comparing the second term with the first term and exchanging them if and only if the second term is smaller, then comparing the third term with the current second term and exchanging them if and only if the third term is smaller, and so on, in order, through comparing the last term, r n , with its current predecessor and exchanging them if and only if the last term is smaller. Figure 3.1 shows how the sequence 1, 9, 8, 7 is transformed into the sequence 1, 8, 7, 9 by one bubble pass.

Calculating in Two Ways: Fubini’s Principle

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.

Addition or Multiplication

Counting is one of the most fundamental skills. People start to count on their fingers when they are in kindergarten or even earlier. But how to count quickly, correctly, and systematically is a lifelong course.