The greatest possible number of points of intersection of 8 straight lines and 4 circles is

Options

A. 104
B. 76
C. 64
D. 32

Answer

To find the greatest possible number of points of intersection of 8 straight lines and 4 circles, we will consider the different types of intersections that can occur: line-to-line, circle-to-circle, and line-to-circle intersections.

### Step-by-Step Solution:

1. **Line-to-Line Intersections**:
- The maximum number of intersection points between straight lines occurs when no two lines are parallel and no three lines are concurrent.
- For \( n \) lines, the maximum number of intersection points is given by the combination formula \( \binom{n}{2} \).
- For 8 lines, the number of intersection points is:
\[ \text{Points from lines} = \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28 \]

2. **Circle-to-Circle Intersections**:
- The maximum number of intersection points between circles occurs when no two circles are tangent and no three circles intersect at the same point.
- For \( m \) circles, the maximum number of intersection points is given by \( 2 \times \binom{m}{2} \) since each pair of circles can intersect at most at 2 points.
- For 4 circles, the number of intersection points is:
\[ \text{Points from circles} = 2 \times \binom{4}{2} = 2 \times \frac{4 \times 3}{2 \times 1} = 2 \times 6 = 12 \]

3. **Line-to-Circle Intersections**:
- Each line can intersect a circle at most at 2 points.
- Therefore, for \( n \) lines and \( m \) circles, the maximum number of intersection points is given by \( 2 \times n \times m \).
- For 8 lines and 4 circles, the number of intersection points is:
\[ \text{Points from lines and circles} = 2 \times 8 \times 4 = 64 \]

4. **Total Points of Intersection**:
- Now, we can sum up all the points of intersection from the three cases:
\[ \text{Total Points} = \text{Points from lines} + \text{Points from circles} + \text{Points from lines and circles} \]
\[ \text{Total Points} = 28 + 12 + 64 = 104 \]

Thus, the greatest possible number of points of intersection of 8 straight lines and 4 circles is **104**.