Which of the following is not a possible number of region into which three straight lines, of infinite extent, divide a plane?

Options

A. 4
B. 5
C. 6
D. 7

Answer

To solve the problem of determining which of the following is not a possible number of regions into which three straight lines of infinite extent can divide a plane, we can follow these steps:

### Step 1: Understand the Problem
We need to find out how many regions can be formed by drawing three straight lines in a plane. We will analyze different configurations of these lines.

### Step 2: Draw the Lines
1. **First Line**: When we draw the first line, it divides the plane into 2 regions.
- **Regions**: 2

2. **Second Line**: When we draw the second line, it can intersect the first line. If it does, it divides the plane into 4 regions.
- **Regions**: 4

3. **Third Line**: When we draw the third line, it can intersect both of the previous lines. If it intersects both, it can create up to 7 regions.
- **Regions**: 7

### Step 3: Count the Regions
From the above configurations, we can summarize the maximum number of regions formed by three lines:
- With 1 line: 2 regions
- With 2 lines: 4 regions
- With 3 lines: 7 regions

### Step 4: Analyze the Options
Now, we need to check the options provided in the question. The possible numbers of regions formed by three lines are 2, 4, 5, 6, and 7.

### Step 5: Identify the Incorrect Option
From our analysis:
- We can achieve 4, 5, 6, and 7 regions with different configurations of the lines.
- However, we cannot achieve exactly 5 or 6 regions with three lines without having the lines parallel or overlapping in a way that does not create distinct regions.

Thus, the number that is not possible is **5**.

### Conclusion
Therefore, the answer to the question is that **5** is not a possible number of regions into which three straight lines can divide a plane.

---