How many 10 digit numbers can be written by using the digits 1 and 2.

Options

A. `""^10 C_1+"^9 C_2`
B. `2^10`
C. `"10 C_2`
D. `10!`

Answer

To determine how many 10-digit numbers can be formed using the digits 1 and 2, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to create a 10-digit number using only the digits 1 and 2. Each position in the number can either be a 1 or a 2.

2. **Identifying Choices for Each Digit**: For each of the 10 positions in the number, we have 2 choices (either 1 or 2).

3. **Calculating Total Combinations**: Since each of the 10 positions can be filled independently with either 1 or 2, we can use the multiplication principle of counting. The total number of combinations can be calculated as:
\[ \text{Total combinations} = 2^{10} \]

4. **Calculating the Value**: Now we calculate \(2^{10}\):
\[ 2^{10} = 1024 \]

5. **Conclusion**: Therefore, the total number of 10-digit numbers that can be formed using the digits 1 and 2 is 1024.

### Final Answer:
The total number of 10-digit numbers that can be written using the digits 1 and 2 is **1024**.

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