10 straight lines, no two of which are parallel and no three of which pass through any common point, are drawn on a plane. The total number of regions (including finite and infinite regions) into which the plane would be divided by the lines is

Options

A. 56
B. 255
C. 1024
D. not unique

Answer

To find the total number of regions into which the plane is divided by 10 straight lines, we can use the formula for the maximum number of regions (R) created by m lines, given by:

\[ R(m) = \frac{m(m + 1)}{2} + 1 \]

### Step-by-step Solution:

1. **Identify the number of lines (m)**:
We have \( m = 10 \) straight lines.

2. **Substitute m into the formula**:
We will substitute \( m = 10 \) into the formula:
\[ R(10) = \frac{10(10 + 1)}{2} + 1 \]

3. **Calculate \( 10 + 1 \)**:
\[ 10 + 1 = 11 \]

4. **Multiply \( 10 \) by \( 11 \)**:
\[ 10 \times 11 = 110 \]

5. **Divide by \( 2 \)**:
\[ \frac{110}{2} = 55 \]

6. **Add \( 1 \)**:
\[ 55 + 1 = 56 \]

7. **Conclusion**:
Therefore, the total number of regions into which the plane is divided by 10 lines is \( 56 \).

### Final Answer:
The total number of regions is \( 56 \).

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