If repetition of digits is allowed, how many three digit even natural numbers can be formed in total?

Introduction / Context:
This question tests understanding of counting principles for number formation. We want to count the number of three digit even natural numbers when digits can be repeated. The even condition restricts the last digit, while the three digit condition restricts the first digit.

Given Data / Assumptions:

• Digits available: 0 to 9.
• We are forming three digit natural numbers.
• The numbers must be even.
• Repetition of digits is allowed.

Concept / Approach:
A three digit number has a hundreds, tens and units place. For a valid three digit number, the hundreds digit cannot be 0. For an even number, the units digit must be one of the even digits 0, 2, 4, 6 or 8. The tens digit can be any digit from 0 to 9 because repetition is allowed.

Step-by-Step Solution:
Step 1: Determine the choices for the units digit (last digit). To be even, it must be one of 0, 2, 4, 6 or 8. Step 2: The number of choices for the units digit is 5. Step 3: The hundreds digit (first digit) cannot be 0, so it can be any of 1, 2, 3, 4, 5, 6, 7, 8 or 9. Step 4: The number of choices for the hundreds digit is 9. Step 5: The tens digit has no restriction other than being a digit from 0 to 9, and repetition of digits is allowed. Step 6: The number of choices for the tens digit is 10. Step 7: Multiply the independent choices: total even three digit numbers = 9 * 10 * 5 = 450.

Verification / Alternative check:
You can reason by cases based on the last digit. For each fixed even last digit, the hundreds digit has 9 choices and the tens digit has 10 choices, giving 9 * 10 = 90 numbers per last digit. With 5 possible last digits, the total is 5 * 90 = 450, which matches the earlier product.

Why Other Options Are Wrong:
500, 540 and 550 correspond to miscounts, often from allowing 0 as a hundreds digit, misunderstanding the even condition, or miscounting the number of even digits available for the last position.

Common Pitfalls:
A major mistake is allowing the first digit to be 0, which leads to two digit or smaller numbers. Another mistake is to think that there are only four even digits by incorrectly excluding 0 from the units place, even though it is allowed for the last position.

Final Answer:
The total number of three digit even natural numbers with repetition allowed is 450.