The number of natural numbers less than 7,000 which can be formed by using the digits 0, 1, 3, 7, 9 (repetition of digits allowed) is equal to

Options

A. 250
B. 374
C. 372
D. 375

Answer

To solve the problem of finding the number of natural numbers less than 7,000 that can be formed using the digits 0, 1, 3, 7, and 9 (with repetition of digits allowed), we will categorize the numbers based on their digit count: 1-digit, 2-digit, 3-digit, and 4-digit numbers.

### Step-by-Step Solution:

1. **1-Digit Numbers:**
- The digits available are 0, 1, 3, 7, and 9.
- Since we are looking for natural numbers, we cannot use 0.
- The valid digits are: 1, 3, 7, 9.
- Therefore, there are **4 one-digit numbers**.

**Hint:** Count only the non-zero digits for natural numbers.

2. **2-Digit Numbers:**
- The first digit cannot be 0 (to ensure it's a 2-digit number).
- Valid choices for the first digit: 1, 3, 7, 9 (4 options).
- The second digit can be any of the 5 digits (0, 1, 3, 7, 9).
- Therefore, the total number of 2-digit numbers is:
\[ 4 \text{ (choices for the first digit)} \times 5 \text{ (choices for the second digit)} = 20 \]

**Hint:** Remember that the first digit cannot be zero.

3. **3-Digit Numbers:**
- Again, the first digit cannot be 0.
- Valid choices for the first digit: 1, 3, 7, 9 (4 options).
- The second and third digits can be any of the 5 digits.
- Therefore, the total number of 3-digit numbers is:
\[ 4 \text{ (choices for the first digit)} \times 5 \text{ (choices for the second digit)} \times 5 \text{ (choices for the third digit)} = 100 \]

**Hint:** Use the same logic for the second and third digits as you did for the second digit in the 2-digit numbers.

4. **4-Digit Numbers (less than 7000):**
- The first digit can only be 1, 3 (it cannot be 7 or 9 as that would make the number 7000 or greater).
- Valid choices for the first digit: 1, 3 (2 options).
- The second, third, and fourth digits can be any of the 5 digits.
- Therefore, the total number of 4-digit numbers is:
\[ 2 \text{ (choices for the first digit)} \times 5 \text{ (choices for the second digit)} \times 5 \text{ (choices for the third digit)} \times 5 \text{ (choices for the fourth digit)} = 250 \]

**Hint:** Ensure the first digit is less than 7 to keep the number under 7000.

5. **Total Count:**
- Now, we add all the numbers we calculated:
\[ \text{Total} = \text{1-digit numbers} + \text{2-digit numbers} + \text{3-digit numbers} + \text{4-digit numbers} \]
\[ \text{Total} = 4 + 20 + 100 + 250 = 374 \]

### Final Answer:
The total number of natural numbers less than 7,000 that can be formed using the digits 0, 1, 3, 7, and 9 is **374**.