To appreciate the value of the inclusion-exclusion principle mentioned in note 4 of the introduction, consider the following problem from the excellent textbook by Saeed Ghahramani, Fundamentals of Probability, (Upper Saddle River, N.J: Prentice-Hall, 1996, p. 69):
From a faculty of six professors, six associate professors, 10 assistant assistant professors, and 12 instructors, a committee of size 6 is formed randomly. What is the probability that there is at least one person from each rank on the committee?
Hint: Be careful, the answer is not
Professor Ghahramani then suggests ...