Solve the following problems independently of each other
If 6 straight lines be drawn in a plane such that no two of them being parallel and no three of them being concurrent, the maximum number of regions into which this plane will be divided

Options

A. 30
B. 22
C. 18
D. 15

Answer

To solve the problem of finding the maximum number of regions into which a plane can be divided by 6 straight lines, given that no two lines are parallel and no three lines are concurrent, we can use a standard formula.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to determine how many regions a plane can be divided into by drawing 6 lines under the given conditions.

2. **Identify the Formula**: The formula to calculate the maximum number of regions (R) created by n lines is:
\[ R(n) = \frac{n(n + 1)}{2} + 1 \]
where \( n \) is the number of lines.

3. **Substituting the Value of n**: In this case, we have \( n = 6 \). We will substitute this value into the formula.
\[ R(6) = \frac{6(6 + 1)}{2} + 1 \]

4. **Calculating the Expression**:
- First, calculate \( 6 + 1 \):
\[ 6 + 1 = 7 \]
- Next, multiply \( 6 \) by \( 7 \):
\[ 6 \times 7 = 42 \]
- Now, divide \( 42 \) by \( 2 \):
\[ \frac{42}{2} = 21 \]
- Finally, add \( 1 \):
\[ 21 + 1 = 22 \]

5. **Conclusion**: The maximum number of regions into which the plane can be divided by 6 lines is **22**.

### Final Answer:
The maximum number of regions into which the plane will be divided is **22**.

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