Key Concepts and Formulas

• Fundamental Principle of Counting (Multiplication Rule): If one event can occur in $m$ ways and another event can occur in $n$ ways, then the two events together can occur in $m \times n$ ways.
• Understanding Number Ranges: Carefully interpreting inequalities like "greater than" and "less than" to determine inclusive or exclusive bounds.
• Digit Place Values: Understanding that in a number $ABCD$, $A$ represents the thousands place, $B$ the hundreds, $C$ the tens, and $D$ the units.

Step-by-Step Solution

Step 1: Define the Problem

We need to find the number of integers between 1000 and 4000 (exclusive) that can be formed using the digits 0, 1, 2, 3, and 4, with repetition allowed. This means we are forming 4-digit numbers.

Step 2: Represent the 4-digit Number

Let the 4-digit number be represented as $ABCD$, where $A$ is the thousands digit, $B$ is the hundreds digit, $C$ is the tens digit, and $D$ is the units digit.

Step 3: Determine the Constraints on the Thousands Digit ($A$)

Since the number must be greater than 1000 and less than 4000, the thousands digit ($A$) can only be 1, 2, or 3. Thus, $A \in \{1, 2, 3\}$.

Step 4: Determine the Constraints on the Hundreds, Tens, and Units Digits ($B, C, D$)

Since repetition is allowed and we can use the digits 0, 1, 2, 3, and 4, each of the digits $B$, $C$, and $D$ can be any of these 5 digits. Thus, $B, C, D \in \{0, 1, 2, 3, 4\}$.

Step 5: Calculate the Number of Possibilities

Since there are 3 choices for the thousands digit ($A$) and 5 choices for each of the hundreds ($B$), tens ($C$), and units ($D$) digits, the total number of such numbers is: $3 \times 5 \times 5 \times 5 = 3 \times 5^3 = 3 \times 125 = 375$

Step 6: Account for the Exclusive Range

We need numbers strictly greater than 1000 and strictly less than 4000. The numbers we counted range from 1000 to 3999. However, the question states "greater than 1000", so we must exclude 1000. The smallest number we can make is 1000 (when $A=1$, $B=0$, $C=0$, $D=0$). Since we are looking for numbers greater than 1000, we must subtract this case.

Therefore, the total number of such numbers is $375 - 1 = 374$.

Common Mistakes & Tips

• Inclusive vs. Exclusive: Carefully distinguish between "greater than or equal to" and "greater than" (similarly for "less than"). This affects whether boundary values are included.
• Zero in the Leading Digit: Remember that the leading digit of a number cannot be zero.

Summary

By considering the constraints on each digit and applying the multiplication principle, we found that there are 375 possible numbers. However, since the question asked for numbers greater than 1000, we excluded 1000 itself, resulting in a final count of 374.

The final answer is $\boxed{374}$, which corresponds to option (C).