What is the MAXIMUM number of points of intersection that can be formed by $5$ straight lines and $3$ circles in a 2D plane, assuming no two straight lines are parallel, no two circles are concentric, and no three lines/circles pass through the same point? ________ (answer in integer).
Answer Key:
46
Step-by-Step Solution & Explanation
**Method 1: Combinatorial Intersection Count**
The total maximum intersections is the sum of three independent pairwise interactions:
1. Line-Line intersections: Each pair of non-parallel lines intersects in at most 1 point:
$I_{LL} = \binom{5}{2} = \frac{5 \times 4}{2} = 10$
2. Circle-Circle intersections: Each pair of non-concentric circles intersects in at most 2 points:
$I_{CC} = \binom{3}{2} \times 2 = 3 \times 2 = 6$
3. Line-Circle intersections: Each line intersects each circle in at most 2 points:
$I_{LC} = 5 \times 3 \times 2 = 30$
$\text{Total Maximum Intersections} = 10 + 6 + 30 = 46$
Integer answer: 46.