Counting Four-Digit Numbers with 3, 4, or Both

Understanding the count of four-digit numbers that include the digits 3, 4, or both is a fascinating mathematical exercise. This article will break down the problem into manageable steps and provide a comprehensive solution.

Introduction to Four-Digit Number Combinations

A four-digit number can be represented as ABCD. Here, each digit can range from 0 to 9. However, for the case of the first digit A, it must not be 0, as that would make the number a three-digit number. Hence, A can range from 1 to 9.

Calculating Total Four-Digit Numbers

The total number of four-digit numbers is straightforward to calculate:

The first digit A has 9 possible values (1, 2, 3, 4, 5, 6, 7, 8, 9). The second digit B has 10 possible values (0 through 9). The third digit C has 10 possible values (0 through 9). The fourth digit D has 10 possible values (0 through 9).

Thus, the total number of four-digit numbers is:

9 x 10 x 10 x 10 9000

Excluding Numbers Without 3 or 4

To calculate the number of four-digit numbers that do not contain the digits 3 or 4, we follow a similar logic:

The first digit A can be any of 1, 2, 5, 6, 7, 8, 9, giving us 7 possibilities. The second digit B can be any of 0, 1, 2, 5, 6, 7, 8, 9, giving us 8 possibilities. The third digit C can be any of 0, 1, 2, 5, 6, 7, 8, 9, giving us 8 possibilities. The fourth digit D can be any of 0, 1, 2, 5, 6, 7, 8, 9, giving us 8 possibilities.

Multiplying these possibilities together, we get:

7 x 8 x 8 x 8 3584

Finding Numbers with 3 or 4

Now, to find the number of four-digit numbers that contain at least one 3 or 4, we subtract the number of four-digit numbers without 3 or 4 from the total number of four-digit numbers:

9000 - 3584 5416

Conclusion

Therefore, the number of four-digit numbers that contain the digits 3 or 4 (or both) is 5416. This calculation can be useful in various applications, such as statistics, probability theory, and combinatorics.

The process of exclusion and inclusion can be applied to other similar problems, making it a valuable tool in understanding combinations and permutations.