How Many Three-Digit Numbers are There with Exactly Three of the Same Digits?

The problem of counting three-digit numbers with exactly three of the same digits is an interesting exploration in combinatorics. Let's delve into a detailed analysis of how to determine such numbers.

Three-Digit Numbers with Three of the Same Digits

The three-digit numbers where all three digits are the same are quite straightforward:

111 222 333 444 555 666 777 888 999

Thus, there are 9 such numbers between 100 to 999.

Four-Digit Numbers with Exactly Three of the Same Digits

For four-digit numbers, the solution involves a more complex calculation. We can break down the process as follows:

Starting with 1

The first set of four-digit numbers with exactly three of the same digits starting with 1 is:

1110 1011

Here, the digit 1 can be replaced by any of the digits from 2 to 9 for the last two digits. Therefore, the total number of such numbers is calculated as:

There are 9 choices for the last digit (2, 3, 4, 5, 6, 7, 8, 9), and for each of these choices, the digits can be arranged in 3! / 2! 3 ways. This gives us 3 * 9 27 numbers.

For Each Digit 2 to 9

A similar approach can be applied for each digit from 2 to 9. Each of these will also generate 27 such numbers. Therefore, the total number of such four-digit numbers is:

9 * 27 243

Total Count from 100 to 10000

Combining the three-digit and four-digit numbers, the total count of numbers from 100 to 10000 with exactly three of the same digits can be calculated as:

9 (three-digit numbers) 243 (four-digit numbers) 252

However, this count can be further verified using a brute force approach with the J programming language:

m.n~./ea 3 ea 1{ ea hist ea sep ea n.100 to 10000 333

The J programming code confirms a total of 333 such numbers. The brute force listing of these numbers is provided above:

111 222 ... (omitted for brevity) 999 1000 1011 1101 1110 ... (remaining numbers)

Conclusion

The problem of counting three-digit or four-digit numbers with exactly three of the same digits is a good exercise in combinatorics. Through logical deduction and programming verification, the total count can be accurately determined. Understanding these principles can be valuable in various fields of mathematics and computer science.