How Many 5-Digit Numbers Can Be Formed Without Repetition from Given Digits?

When faced with the task of forming a 5-digit number from a set of given digits, one must adhere to specific rules to ensure that the number remains valid. In this article, we'll delve into how many 5-digit numbers can be formed from the digits 0, 1, 4, 6, 7, and 9 without repetition, a common problem in permutations and combinations.

Understanding the Problem

The problem statement requires us to form 5-digit numbers using the digits 0, 1, 4, 6, 7, and 9 without repeating any digit. A key aspect to consider is that the first digit of any 5-digit number cannot be 0, as such a configuration would not meet the 5-digit criteria.

Steps to Calculate

Choosing the First Digit

For the first digit, we need to select from the digits 1, 4, 6, 7, and 9 since 0 cannot be the first digit. This gives us 5 possible choices for the first digit.

Choosing the Remaining Digits

After selecting the first digit, we are left with 5 digits (including 0) to choose from for the remaining 4 digits. We need to find the number of permutations of these 5 digits taken 4 at a time, effectively forming the rest of the 5-digit number.

Permutations Calculation

The total number of permutations of the remaining 4 digits can be calculated using the formula for permutations, which is Pnr n! / (n-r)! . In this case, the total permutations of the remaining 4 digits from 5 available digits is:

P54 5! / (5-4)! 5! / 1! 5 * 4 * 3 * 2 120

Total Arrangements

To find the total number of 5-digit numbers, we multiply the number of choices for the first digit by the number of permutations of the remaining digits:

120 * 5 600

Conclusion

By following the steps outlined above, we conclude that there are 600 unique 5-digit numbers that can be formed from the digits 0, 1, 4, 6, 7, and 9 without repeating any digit. This calculation is crucial for understanding permutations in digit selection problems.

Additional Examples

6-Digit Numbers from Digits 1, 3, 4, 6, 7, and 9

For another example, if we want to form 5-digit numbers from the digits 1, 3, 4, 6, 7, and 9, the number of possible permutations is:

P65 6! / (6-5)! 6! / 1! 6 * 5 * 4 * 3 * 2 720

8-Digit Numbers

For a similar problem with 8 digits, the number of permutations of 5 digits from 8 is:

P85 8! / (8-5)! 8! / 3! 8 * 7 * 6 * 5 * 4 6,720

Summary

In conclusion, understanding how many 5-digit numbers can be formed without repetition is a fundamental concept in combinatorics, crucial for various mathematical and real-world applications. By following the steps outlined, we can efficiently determine the number of unique combinations possible with any set of digits under given constraints.