Number of natural numbers `lt2.10^(4)`, which can be formed with the digits, 1,2,3 only is equal to

Options

A. `(3^(5).2.3^(4)-3)/2`
B. `(3^(5)-2.3^(4)+3)/2`
C. `(3^(7)-1)/2`
D. None of these

Answer

To find the number of natural numbers less than \(2 \times 10^4\) that can be formed using the digits 1, 2, and 3, we will consider different cases based on the number of digits in the natural numbers.

### Step-by-Step Solution:

1. **Understanding the Range**:
We need to find natural numbers less than \(2 \times 10^4 = 20000\). This means we can have 1-digit, 2-digit, 3-digit, and 4-digit numbers. We cannot have 5-digit numbers since the smallest 5-digit number is \(10000\) and the largest is \(99999\), which exceeds \(20000\).

2. **Case 1: 1-digit Numbers**:
The possible 1-digit numbers are 1, 2, and 3.
- Total 1-digit numbers = 3.

3. **Case 2: 2-digit Numbers**:
Each digit can be 1, 2, or 3.
- For the first digit, we have 3 choices (1, 2, or 3).
- For the second digit, we also have 3 choices.
- Total 2-digit numbers = \(3 \times 3 = 9\).

4. **Case 3: 3-digit Numbers**:
Similar to the 2-digit case:
- For the first digit, we have 3 choices.
- For the second digit, we have 3 choices.
- For the third digit, we have 3 choices.
- Total 3-digit numbers = \(3 \times 3 \times 3 = 27\).

5. **Case 4: 4-digit Numbers**:
Again, each digit can be 1, 2, or 3:
- For the first digit, we have 3 choices.
- For the second digit, we have 3 choices.
- For the third digit, we have 3 choices.
- For the fourth digit, we have 3 choices.
- Total 4-digit numbers = \(3 \times 3 \times 3 \times 3 = 81\).

6. **Total Count of Natural Numbers**:
Now, we sum all the possibilities from each case:
\[ \text{Total} = \text{(1-digit)} + \text{(2-digit)} + \text{(3-digit)} + \text{(4-digit)} \]
\[ \text{Total} = 3 + 9 + 27 + 81 = 120. \]

7. **Conclusion**:
The total number of natural numbers less than \(2 \times 10^4\) that can be formed using the digits 1, 2, and 3 is **120**.