Using the digits 0, 1, 2, 3 and 4, how many integers greater than 999 but not greater than 4000 can be formed, if repetition of digits is allowed?

Introduction / Context:
This question focuses on counting 4 digit integers under specific constraints. The digits that can be used are limited, repetition is allowed, and there is a bound on how large the number can be. It tests careful case analysis and understanding of place value constraints.

Given Data / Assumptions:

• Allowed digits: 0, 1, 2, 3, 4.
• Integers must be greater than 999 and not greater than 4000.
• Repetition of digits is allowed.
• We are effectively dealing with 4 digit numbers between 1000 and 4000 inclusive.

Concept / Approach:
All required numbers have 4 digits. Let the digits be d1 d2 d3 d4. The leading digit d1 cannot be 0, or else the number would not be 4 digit. There is also an upper bound: the number must not exceed 4000. We split into cases based on d1:

• Case 1: d1 = 1, 2 or 3.
• Case 2: d1 = 4, but the number cannot exceed 4000.

Step-by-Step Solution:
Step 1: For 4 digit numbers using digits 0 to 4, d1 cannot be 0. Step 2: Consider Case 1: d1 can be 1, 2 or 3. That gives 3 choices for d1. Step 3: For each such choice, digits d2, d3 and d4 can each be any of the 5 digits 0, 1, 2, 3, 4 (repetition allowed). Step 4: So number of possibilities for Case 1 = 3 * 5 * 5 * 5 = 3 * 125 = 375. Step 5: Consider Case 2: d1 = 4. The maximum allowed number is 4000, so the remaining digits must be 000; any larger combination like 4100 would exceed 4000. Step 6: Therefore, the only valid number with leading digit 4 is 4000 itself. Step 7: Count for Case 2 = 1. Step 8: Total number of valid integers = 375 + 1 = 376.

Verification / Alternative check:
We can confirm the logic by understanding that all 4 digit numbers from 1000 to 3999 are automatically valid as long as their digits are restricted to 0 to 4, which is handled in Case 1 with 375 combinations. For numbers starting with 4, only 4000 qualifies because any other choice for the remaining digits would create a value larger than 4000. This reasoning aligns with the numeric counts computed earlier.

Why Other Options Are Wrong:
375: This omits the extra valid number 4000 from Case 2. 500 and 673: These overcount and would require including invalid leading digits or numbers above 4000. 250: This undercounts and would likely result from miscounting possibilities or forgetting that repetition is allowed.

Common Pitfalls:
A very common mistake is to ignore the upper bound of 4000 and simply count all 4 digit numbers with allowed digits, or to forget that 0 cannot appear as the first digit. Another pitfall is failing to treat the leading digit 4 as a separate case and accidentally including many numbers from 4001 to 4444, which exceed the given limit. Systematic case analysis helps avoid these issues.

Final Answer:
The number of integers greater than 999 but not greater than 4000 that can be formed is 376.