Determining the number of four-digit numbers without repeated digits involves understanding the concept of permutations and applying it to the specific constraints of the problem. This process requires considering the available choices for each digit and ensuring no repetition occurs. Let's delve into the methodology and arrive at the solution.
Finding the Number of 4-Digit Numbers with No Repeated Digits
To find the number of four-digit numbers with no repeated digits, we need to consider the available options for each digit.
Understanding the Constraints
- Four-Digit Number: The number must have four digits, meaning it must be between 1000 and 9999.
- No Repeated Digits: Each digit in the number must be unique.
The Permutation Approach
We can solve this problem using permutations. A permutation is an arrangement of objects in a specific order. In this case, we are arranging four digits, and the order matters.
- Thousands Place: We have 9 choices for the thousands digit (1 through 9). We cannot use 0 as it would make the number a three-digit number.
- Hundreds Place: After choosing the thousands digit, we have 9 remaining choices for the hundreds digit.
- Tens Place: We are left with 8 choices for the tens digit.
- Units Place: Finally, we have 7 choices for the units digit.
Calculation
To find the total number of possible arrangements, we multiply the number of choices for each digit:
9 * 9 * 8 * 7 = 4536
Therefore, there are 4536 four-digit numbers with no repeated digits.
Example
Let's look at an example to illustrate this:
- Thousands Digit: Let's choose 2.
- Hundreds Digit: We can choose any digit except 2, so let's choose 7.
- Tens Digit: We can choose any digit except 2 or 7, so let's choose 5.
- Units Digit: We can choose any digit except 2, 7, or 5, so let's choose 1.
This gives us the number 2751. We can follow this process with different choices for each digit to get all 4536 possible numbers.
Conclusion
The number of four-digit numbers with no repeated digits is 4536. This was determined by considering the available choices for each digit and applying the principles of permutations. Understanding permutations is crucial for solving problems that involve arranging objects in a specific order, particularly when restrictions like no repeated digits are involved. By carefully analyzing the constraints and applying the appropriate mathematical tools, we can arrive at the solution.
