How many 10 digits numbers can be formed by using the digits 2 and 3?

Options

A. `2^(10)`
B. `10^2`
C. `10!`
D. none of these

Answer

To determine how many 10-digit numbers can be formed using the digits 2 and 3, we can follow these steps:

### Step 1: Understand the problem
We need to create a 10-digit number using only the digits 2 and 3. Each digit in the number can independently be either 2 or 3.

### Step 2: Determine the choices for each digit
For each of the 10 positions in the number, we have 2 choices (either 2 or 3).

### Step 3: Calculate the total combinations
Since the choice for each digit is independent of the others, we can use the multiplication principle of counting. The total number of combinations can be calculated as follows:

- For the first digit, we have 2 choices (2 or 3).
- For the second digit, we also have 2 choices (2 or 3).
- This continues for all 10 digits.

Thus, the total number of 10-digit combinations can be expressed mathematically as:
\[ \text{Total combinations} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^{10} \]

### Step 4: Calculate \(2^{10}\)
Now, we calculate \(2^{10}\):
\[ 2^{10} = 1024 \]

### Conclusion
Therefore, the total number of 10-digit numbers that can be formed using the digits 2 and 3 is **1024**.

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