Counting 5-Digit Numbers Containing at Least One '5'

The problem of counting 5-digit numbers that contain at least one '5' is a classic example in combinatorics. It engages the concept of permutations with repetitions and provides a good exercise in understanding how to exclude unwanted cases from a large set.

Introduction to the Problem

First, let's consider the total number of 5-digit numbers. A 5-digit number ranges from 10000 to 99999. Each digit can be any of the 10 digits (0-9). However, the first digit cannot be 0. Thus, the total number of 5-digit numbers is calculated as follows:

Total Permutations with Repetitions

(9 * 10^4)  90,000

Excluding Numbers Without '5'

To count the numbers containing at least one '5', it is easier to first count the numbers that do not contain any '5' and then subtract this from the total number of 5-digit numbers. If a number does not contain '5', then each digit can be any of the other 9 digits (0-9 excluding 5).

Permutations Without '5'

(9^5)  59,049

Note that for the first digit, we still exclude 0 to ensure it remains a 5-digit number. However, since 5 is also excluded, we take the much simpler approach of just excluding 5 for this problem.

Applying the Inclusion-Exclusion Principle

Thus, the number of 5-digit numbers that contain at least one '5' is the total number of 5-digit numbers minus the number of 5-digit numbers that do not contain any '5'.

Calculation

(90,000 - 59,049)  30,951

Conclusion

Therefore, the number of 5-digit numbers that contain at least one '5' is 30,951.

Further Exploration

This problem can be generalized to understand permutations with repetitions and inclusion-exclusion principle. It can be applied to various other combinatorial problems, such as counting numbers with specific digit properties or patterns.

Additional Tips

Be mindful of constraints: Always consider any constraints, like the first digit not being 0, when counting permutations. Use inclusion-exclusion method: This method helps in systematically calculating the number of cases that meet certain criteria by adding and subtracting overlapping sets. Practicing variations: Try solving similar problems with different digit constraints to improve your understanding and problem-solving skills.

References:

[1]

[2] _principle

By applying these concepts, you can solve a wide range of combinatorial problems in a structured and efficient manner.