## To prove that there are at least three parts with the same serial number, we can apply the Pigeonhole Principle.

Step 1: Understand the Problem

The problem states that there is a total of 60,000 parts produced in the factory, and each part has a unique serial number consisting of a letter and three digits.

Step 2: Determine the Number of Possible Serial Numbers

To find the number of possible serial numbers, we need to consider the number of options for each digit in the serial number.

For the letter, there are 26 options (A-Z).

For each digit, there are 10 options (0-9).

Therefore, the total number of possible serial numbers is 26 × 10 × 10 × 10 = 26,000.

Step 3: Divide the Total Number of Parts by the Number of Possible Serial Numbers

Dividing the total number of parts (60,000) by the number of possible serial numbers (26,000) will give us the average number of parts per serial number.

60,000 ÷ 26,000 ≈ 2.31

Since the average number is 2.31, we can conclude that on average, each serial number will be assigned to slightly more than 2 parts.

Step 4: Apply the Pigeonhole Principle

The Pigeonhole Principle states that if you have more pigeons than pigeonholes, at least one pigeonhole must contain multiple pigeons.

In this case, the "pigeons" are the serial numbers (60,000 parts), and the "pigeonholes" are the possible serial numbers (26,000).

Since there are more parts (pigeons) than possible serial numbers (pigeonholes), at least one serial number (pigeonhole) must contain more than one part (pigeon).

Therefore, there must be at least three parts with the same serial number.

Step 5: Conclusion

By applying the Pigeonhole Principle, we have proven that in a factory that produced 60,000 parts with a total of 26,000 possible serial numbers, there must be at least three parts with the same serial number.