How many different 4 digit numbers greater than 6000 can be formed using the digits 0, 2, 4 and 6, if repetition of digits is allowed in any position?

Introduction / Context:
This problem is about constructing 4 digit numbers from a small set of digits, with repetition permitted, and subject to the condition that the final number must be greater than 6000. It checks understanding of place value, especially the restriction on the leading digit, and the use of simple counting with repetition allowed.

Given Data / Assumptions:

• Available digits are 0, 2, 4 and 6.
• Numbers formed must be 4 digit numbers.
• Digits may repeat in the same number.
• The overall number must be greater than 6000.
• 4 digit numbers cannot start with 0.

Concept / Approach:
The key idea is to treat each place in the 4 digit number as a slot. The thousands place must be chosen carefully so that the resulting number exceeds 6000. Once the thousands digit is fixed, the remaining positions can be filled freely using the allowed digits because repetition is permitted. We use a simple multiplication principle for counting such arrangements.

Step-by-Step Solution:
Step 1: For a 4 digit number to be greater than 6000 using digits 0, 2, 4 and 6, the thousands digit must be 6. If it were 0, 2 or 4 the number would be less than 6000.Step 2: So the thousands digit has exactly 1 choice, which is 6.Step 3: The remaining three positions are the hundreds, tens and units places. Each of these can be filled with any of the 4 digits 0, 2, 4 or 6, because repetition is allowed.Step 4: Therefore the hundreds place has 4 choices, the tens place has 4 choices and the units place has 4 choices.Step 5: Using the multiplication principle, total numbers = 1 * 4 * 4 * 4 = 4^3 = 64.

Verification / Alternative check:

Why Other Options Are Wrong:

• 63 and 62: These are close to 64 but suggest that the student has subtracted one or two numbers incorrectly, possibly due to mistaken exclusion of numbers like 6000 or confusion about the leading digit condition.
• 60: This undercounts by not allowing all combinations of the inner digits or by assuming that 0 cannot appear in positions other than the thousands place, which is incorrect.

Common Pitfalls:
One common error is to allow the thousands digit to be 0, 2 or 4, which would produce numbers less than or equal to 6000 and violate the condition. Another frequent mistake is to think that 0 cannot appear in any position, whereas the restriction only applies to the leading digit in a positive integer. Learners also sometimes forget that repetition is allowed, and they incorrectly use permutations without repetition.

Final Answer:
The number of different 4 digit numbers greater than 6000 that can be formed is 64.