The maximum number of points of intersection into which 4 circles and 4 straight lines intersect, is

Options

A. 26
B. 50
C. 56
D. 72

Answer

To find the maximum number of points of intersection formed by 4 circles and 4 straight lines, we can analyze the intersections in three categories: line-line intersections, line-circle intersections, and circle-circle intersections.

### Step-by-Step Solution:

1. **Line-Line Intersections**:
- The maximum number of intersection points between straight lines can be calculated using the combination formula \( nC2 \), where \( n \) is the number of lines.
- For 4 lines, the number of intersection points is given by:
\[ \text{Line-Line Intersections} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

2. **Line-Circle Intersections**:
- Each line can intersect a circle at most at 2 points. Therefore, for 4 lines and 4 circles, the maximum number of intersection points is:
\[ \text{Line-Circle Intersections} = 4 \text{ lines} \times 4 \text{ circles} \times 2 \text{ points} = 4 \times 4 \times 2 = 32 \]

3. **Circle-Circle Intersections**:
- The maximum number of intersection points between circles can also be calculated using the combination formula, but since each pair of circles can intersect at most at 2 points, we have:
\[ \text{Circle-Circle Intersections} = \binom{4}{2} \times 2 = \frac{4!}{2!(4-2)!} \times 2 = 6 \times 2 = 12 \]

4. **Total Maximum Intersections**:
- Now, we sum all the intersection points calculated from the three categories:
\[ \text{Total Intersections} = \text{Line-Line Intersections} + \text{Line-Circle Intersections} + \text{Circle-Circle Intersections} \]
\[ \text{Total Intersections} = 6 + 32 + 12 = 50 \]

Thus, the maximum number of points of intersection into which 4 circles and 4 straight lines intersect is **50**.