Maximum number of points into which 3 circles and 3 lines intersect is :

Options

A. 21
B. 9
C. 27
D. 3!

Answer

To find the maximum number of points into which 3 circles and 3 lines can intersect, we can break the problem down into several parts:

### Step 1: Calculate the intersection points among the lines
Three lines can intersect each other at most at 3 points. The number of intersection points formed by \( n \) lines is given by the combination formula \( \binom{n}{2} \), which represents the number of ways to choose 2 lines from \( n \) lines.

\[ \text{Intersection points from lines} = \binom{3}{2} = 3 \]

### Step 2: Calculate the intersection points among the circles
Next, we consider the circles. Each pair of circles can intersect at most at 2 points. For 3 circles, the number of intersection points is given by \( \binom{n}{2} \) multiplied by 2 (since each pair intersects at 2 points).

\[ \text{Intersection points from circles} = \binom{3}{2} \times 2 = 3 \times 2 = 6 \]

### Step 3: Calculate the intersection points between lines and circles
Now, we need to consider how the lines intersect with the circles. Each line can intersect each circle at most at 2 points. Therefore, for 3 lines and 3 circles, the total number of intersection points is:

\[ \text{Intersection points between lines and circles} = 3 \text{ lines} \times 3 \text{ circles} \times 2 = 3 \times 3 \times 2 = 18 \]

### Step 4: Total intersection points
Now, we can sum all the intersection points calculated from the lines, circles, and their combinations:

\[ \text{Total points} = \text{Intersection points from lines} + \text{Intersection points from circles} + \text{Intersection points between lines and circles} \]
\[ \text{Total points} = 3 + 6 + 18 = 27 \]

Thus, the maximum number of points into which 3 circles and 3 lines can intersect is **27**.

### Final Answer
The maximum number of points into which 3 circles and 3 lines intersect is **27**.

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