A factory makes automobile parts. Each part has a code consisting of a letter and three digits, such as C117, O076, or Z920. Last week the factory made 60,000 parts. Prove that there are at least three parts that have the same serial number.

There are 26 ways to choose the letter, and 10 ways to choose each digit for a total of 26*10³=26000 distinct serial numbers.

So there are 34000 parts that do not have distinct serial numbers. Out of the 34000 parts, we can take another 26000 to make duplicate serial numbers with the first 26000. This accounts for 52000 parts, all of which are duplicates.

To assign serial numbers to the remaining 8000 parts using the same scheme, we have to reuse at least one of the 26000 serial numbers. Therefore by the pigeon hole principle, at least three parts have the same serial number.