Understanding and Calculating Three-Digit Numbers with All Digits Different

The problem of determining how many three-digit numbers between 100 and 999 have all different digits is a fascinating exercise in number theory and combinatorics. This article not only provides a step-by-step solution but also delves into the logic behind the calculation, illustrating how to arrive at the correct answer using both reasoning and Python code.

Step-by-Step Solution

Let’s break down the process of finding three-digit numbers with all different digits:

Choosing the First Digit (Hundreds Place)

The first digit of a three-digit number must be between 1 and 9 (inclusive) because it cannot be 0. Therefore, there are 9 possible choices for the first digit (1 through 9).

Choosing the Second Digit (Tens Place)

The second digit can be any digit from 0 to 9, but it must be different from the first digit. This means we have 10 total digits from 0 to 9 minus the 1 digit that has already been used as the first digit, leaving us with 9 possible choices.

Choosing the Third Digit (Units Place)

The third digit can also be any digit from 0 to 9, but it must be different from both the first and second digits. Therefore, we have 10 total digits minus the 2 digits that have already been used, resulting in 8 possible choices.

Calculating the Total Number of Possible Three-Digit Numbers

We can calculate the total number of three-digit numbers with all different digits using the multiplication principle, which is a common technique in combinatorics. Here is the calculation:

Total Numbers  Choices for First Digit times; Choices for Second Digit times; Choices for Third Digit

Plugging in the values:

Total Numbers 9 times; 9 times; 8

Performing the multiplication:

9 times; 9 8181 times; 8 648

Thus, the total number of three-digit numbers between 100 and 999 in which all three digits are different is 648.

Verification with Python Code

Here is a Python code snippet that verifies the result by iterating through all three-digit numbers and counting those with all different digits:

count  0for num in range(100, 999):    if len(set(str(num)))  3:        count   1count

The output of the code confirms the result, as it returns 648.

Additional Insight

For completeness, let’s briefly discuss numbers with repeated digits:

The numbers 111, 222, 333, 444, 555, 666, 777, 888, and 999 are the only three-digit numbers in which all digits are the same. There are 9 such numbers (one for each digit from 1 to 9).

The total number of three-digit numbers is 900 (from 100 to 999). Subtracting the 9 numbers with repeated digits, we get:

900 - 9 891

Thus, there are 891 three-digit numbers with at least one repeated digit.

Conclusion

To summarize, the total number of three-digit numbers between 100 and 999 with all different digits is 648, as calculated using a combination of mathematical reasoning and a Python script. This problem not only reinforces basic combinatorial principles but also demonstrates the practical application of coding to verify mathematical results.