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

Options

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

Answer

To find the greatest possible number of points of intersection of 8 straight lines and 4 circles, we will break down the problem into three cases: intersections between lines, intersections between circles, and intersections between lines and circles.

### Step-by-Step Solution:

**Step 1: Intersections between Lines**
- The maximum number of intersection points formed by \( n \) lines is given by the formula:
\[ \text{Number of intersections} = \binom{n}{2} \]
- For 8 lines, we calculate:
\[ \binom{8}{2} = \frac{8 \times 7}{2} = 28 \]

**Step 2: Intersections between Circles**
- The maximum number of intersection points formed by \( m \) circles is given by the formula:
\[ \text{Number of intersections} = \binom{m}{2} \times 2 \]
- For 4 circles, we calculate:
\[ \binom{4}{2} = \frac{4 \times 3}{2} = 6 \]
- Since each pair of circles can intersect at 2 points:
\[ \text{Total intersections from circles} = 6 \times 2 = 12 \]

**Step 3: Intersections between Lines and Circles**
- Each line can intersect with each circle at 2 points. Therefore, the total number of intersections between lines and circles is:
\[ \text{Total intersections} = \text{Number of lines} \times \text{Number of circles} \times 2 \]
- For 8 lines and 4 circles:
\[ \text{Total intersections} = 8 \times 4 \times 2 = 64 \]

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

### Final Answer:
The greatest possible number of points of intersection of 8 straight lines and 4 circles is **104**.

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