How many natural numbers less than 1000 can be formed from the digits 0, 1, 2, 3, 4, 5 when a digit may be repeated any number of times?

Explanation

To solve the problem of how many natural numbers less than 1000 can be formed from the digits 0, 1, 2, 3, 4, and 5 (with repetition allowed), we will consider the numbers based on their digit count: 1-digit, 2-digit, and 3-digit numbers.

### Step-by-step Solution:

1. **Count 1-digit natural numbers:**
- The digits available are 0, 1, 2, 3, 4, and 5.
- However, since we are looking for natural numbers, we cannot include 0.
- Therefore, the possible 1-digit natural numbers are: 1, 2, 3, 4, 5.
- **Total 1-digit numbers = 5.**

2. **Count 2-digit natural numbers:**
- The first digit (tens place) cannot be 0 (to ensure it's a 2-digit number).
- Hence, the first digit can be any of the digits: 1, 2, 3, 4, or 5 (5 options).
- The second digit (units place) can be any of the 6 digits (0, 1, 2, 3, 4, 5).
- Therefore, the total number of 2-digit numbers can be calculated as:
\[ \text{Total 2-digit numbers} = 5 \text{ (choices for the first digit)} \times 6 \text{ (choices for the second digit)} = 30. \]

3. **Count 3-digit natural numbers:**
- The first digit (hundreds place) again cannot be 0.
- So, the first digit can be any of the digits: 1, 2, 3, 4, or 5 (5 options).
- The second digit (tens place) can be any of the 6 digits (0, 1, 2, 3, 4, 5).
- The third digit (units place) can also be any of the 6 digits (0, 1, 2, 3, 4, 5).
- Therefore, the total number of 3-digit numbers can be calculated as:
\[ \text{Total 3-digit numbers} = 5 \text{ (choices for the first digit)} \times 6 \text{ (choices for the second digit)} \times 6 \text{ (choices for the third digit)} = 180. \]

4. **Calculate the total natural numbers less than 1000:**
- Now, we sum up all the counts from the previous steps:
\[ \text{Total natural numbers} = \text{Total 1-digit numbers} + \text{Total 2-digit numbers} + \text{Total 3-digit numbers} = 5 + 30 + 180 = 215. \]

### Final Answer:
The total number of natural numbers less than 1000 that can be formed from the digits 0, 1, 2, 3, 4, and 5 is **215**.

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