3.5 Pigeonhole Principle, Inclusion-Exclusion

Class Schedules

At a certain school, there are 11 different classes offered to first-year students, and each student must enroll in exactly 4 of them. How many students must be in one class year to guarantee that at least 2 students will have the same schedule?
Either input the final result or an equivalent form, using fact(x) to denote \(x!\); explictly write out multiplication with *, and ** for exponentiation.

Exercise 1

This is a pigeonhole problem (although it's also a counting problem). The pigeons are the students, and the pigeonholes are the possible class schedules, of which there are 11 choose 4 = 330. A student is mapped to a schedule if the student has that schedule. To guarantee that 2 students will have the same schedule, there must be more pigeons than holes -- so there must be at least 331 students in a class year.