How many 10-digit numbers can be formed by using digits 1 and 2

Options

A. `""^(10)C_(1)+""^(9)C_(2)`
B. `2^(10)`
C. `10^(2)`
D. 10!

Answer

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

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

**Hint:** Think about how many choices you have for each digit in the number.

### Step 2: Determine the Choices for Each Digit
For each of the 10 positions in the number, you have 2 choices: you can either place a 1 or a 2.

**Hint:** Consider how many positions there are and how many choices you have for each position.

### Step 3: Calculate the Total Combinations
Since there are 10 positions and each position has 2 choices, the total number of combinations can be calculated using the formula for combinations of independent events. This is given by:

\[ \text{Total combinations} = \text{(Number of choices)}^{\text{(Number of positions)}} \]

In this case:

\[ \text{Total combinations} = 2^{10} \]

**Hint:** Remember that when you have multiple independent choices, you multiply the number of choices.

### Step 4: Compute \(2^{10}\)
Now we need to calculate \(2^{10}\):

\[ 2^{10} = 1024 \]

**Hint:** You can calculate \(2^{10}\) by multiplying 2 by itself 10 times or using a calculator.

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

**Final Answer:** 1024