# 3.5 Pigeonhole Principle, Inclusion-Exclusion

## Rolling Dice

You roll a pair of dice 25 times, and each time you record the sum of the numbers on the upward faces. At least how many sums are guaranteed to be the same?

Exercise 1

Let the integers [2, 12] be the holes, and the sum of the numbers on the dices' upward faces be the pigeons. By the Generalized Pigeonhole Principle, some hole has $$\geq \lceil\frac{25}{12-2}\rceil=3$$ pigeons.