Counting Four-Digit Numbers with Even Digits
The question asks us to find the total number of four-digit natural numbers where all the digits used are even.
Understanding the Requirements
A natural number is a positive whole number (1, 2, 3, ...). A four-digit number is a natural number between 1000 and 9999, inclusive.
The digits that are considered even are: 0, 2, 4, 6, 8. There are 5 even digits in total.
Applying Constraints to Digits
Let's consider the four positions of a four-digit number:
- Thousands place
- Hundreds place
- Tens place
- Units place
We need to determine the number of choices for each place, keeping in mind the constraints:
- The number must be a four-digit number, meaning the first digit (thousands place) cannot be 0.
- All digits must be even (from the set {0, 2, 4, 6, 8}).
Calculating Choices for Each Digit
- Thousands place: This digit must be even, but it cannot be 0. So, the possible choices are {2, 4, 6, 8}. This gives us 4 possible choices.
- Hundreds place: This digit can be any of the even digits. The possible choices are {0, 2, 4, 6, 8}. This gives us 5 possible choices.
- Tens place: Similar to the hundreds place, this digit can be any of the even digits {0, 2, 4, 6, 8}. This gives us 5 possible choices.
- Units place: This digit can also be any of the even digits {0, 2, 4, 6, 8}. This gives us 5 possible choices.
Total Count Calculation
To find the total number of such four-digit natural numbers, we multiply the number of choices for each digit together. This is based on the fundamental principle of counting.
Total count = (Choices for Thousands) \(\times\) (Choices for Hundreds) \(\times\) (Choices for Tens) \(\times\) (Choices for Units)
Total count = \(4 \times 5 \times 5 \times 5\)
Total count = \(4 \times (5^3)\)
Total count = \(4 \times 125\)
Total count = 500
Conclusion
Therefore, there are 500 four-digit natural numbers where all the digits are even.
Similar Questions
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Important Questions from Permutation and Combination
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