Key Concepts and Formulas

• Fundamental Principle of Counting (Multiplication Principle): If there are $m$ ways to do one thing and $n$ ways to do another, then there are $m \times n$ ways to do both.
• Natural Numbers: Positive integers (1, 2, 3, ...). Zero is not a natural number.
• Understanding place value (ones, tens, hundreds, thousands, etc.)

Step-by-Step Solution

Step 1: Analyze the Problem and Establish Cases

We need to find the number of natural numbers less than 7000 that can be formed using the digits 0, 1, 3, 7, and 9, with repetition allowed. This means we need to consider 1-digit, 2-digit, 3-digit, and 4-digit numbers separately.

Step 2: Calculate the Number of 1-Digit Numbers

• What: Find the number of 1-digit natural numbers we can form.
• Why: The number must be greater than 0.
• Available Digits: {0, 1, 3, 7, 9}
• Working: We exclude 0. The possible digits are {1, 3, 7, 9}.
• Number of 1-digit numbers: 4

Step 3: Calculate the Number of 2-Digit Numbers

• What: Find the number of 2-digit natural numbers we can form.
• Why: The first digit cannot be 0.
• Available Digits: {0, 1, 3, 7, 9}
• Working: Let the 2-digit number be $AB$. $A$ can be {1, 3, 7, 9} (4 choices). $B$ can be {0, 1, 3, 7, 9} (5 choices).
• Number of 2-digit numbers: $4 \times 5 = 20$

Step 4: Calculate the Number of 3-Digit Numbers

• What: Find the number of 3-digit natural numbers we can form.
• Why: The first digit cannot be 0.
• Available Digits: {0, 1, 3, 7, 9}
• Working: Let the 3-digit number be $ABC$. $A$ can be {1, 3, 7, 9} (4 choices). $B$ can be {0, 1, 3, 7, 9} (5 choices). $C$ can be {0, 1, 3, 7, 9} (5 choices).
• Number of 3-digit numbers: $4 \times 5 \times 5 = 100$

Step 5: Calculate the Number of 4-Digit Numbers Less Than 7000

• What: Find the number of 4-digit natural numbers less than 7000 we can form.
• Why: The first digit cannot be 0 and must be less than 7.
• Available Digits: {0, 1, 3, 7, 9}
• Working: Let the 4-digit number be $ABCD$. $A$ can be {1, 3} (2 choices, since the number must be less than 7000). $B$ can be {0, 1, 3, 7, 9} (5 choices). $C$ can be {0, 1, 3, 7, 9} (5 choices). $D$ can be {0, 1, 3, 7, 9} (5 choices).
• Number of 4-digit numbers: $2 \times 5 \times 5 \times 5 = 250$

Step 6: Calculate the Total Number of Natural Numbers

• What: Sum the number of numbers from each case.
• Why: These are disjoint cases, so we can simply add them.
• Working: Total = (1-digit numbers) + (2-digit numbers) + (3-digit numbers) + (4-digit numbers) = $4 + 20 + 100 + 250 = 374$

Common Mistakes & Tips

• Remember that "natural numbers" do not include 0.
• When forming numbers, the first digit cannot be zero unless it's a 1-digit number.
• Pay close attention to restrictions like "less than 7000." This limits the choices for the leading digit in the 4-digit case.

Summary

We systematically broke down the problem into cases based on the number of digits. We applied the Fundamental Principle of Counting, being careful to account for the restrictions on the leading digit (cannot be 0) and the overall constraint that the number must be less than 7000. The total number of natural numbers less than 7000 that can be formed is 374.

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