Show that if there are 30 students in a class, then at least two have last names that begin with the same...?

Question

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I believe this requires the pigeon hole principle and would like a solution and explanation.

Answer 1

The pigeonhole principle states that, given two natural numbers n and m with n > m, if n items are put into m pigeonholes, then at least one pigeonhole must contain more than one item.

Imagine 26 boxes each labeled with a letter A through Z. If you take each student's name, you can put them in one of the boxes based on the first letter of their last name. But since you have 30 names and 26 boxes, at least one box will have more than one name in it. This is equivalent to saying that at least two students will have names that begin with the same letter.