Find the maximum possible number of regions created by 6 overlapping triangles.

Options

A. 36
B. 63
C. 92
D. 96

Answer

To find the maximum possible number of regions created by 6 overlapping triangles, we can use the combinatorial approach to count the regions formed by the intersections of these triangles.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to determine how many distinct regions can be formed when 6 triangles overlap. Each triangle can intersect with others, creating new regions.

2. **Using Combinations**:
We can think of the problem in terms of combinations. For each triangle, we can consider how many triangles it overlaps with. The maximum regions created by overlapping triangles can be calculated using combinations of the triangles.

3. **Calculate the Combinations**:
We will calculate the combinations for selecting 1 to 6 triangles:
- For 1 triangle: \( \binom{6}{1} \)
- For 2 triangles: \( \binom{6}{2} \)
- For 3 triangles: \( \binom{6}{3} \)
- For 4 triangles: \( \binom{6}{4} \)
- For 5 triangles: \( \binom{6}{5} \)
- For 6 triangles: \( \binom{6}{6} \)

4. **Calculating Each Combination**:
- \( \binom{6}{1} = 6 \)
- \( \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15 \)
- \( \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \)
- \( \binom{6}{4} = \frac{6 \times 5}{2 \times 1} = 15 \)
- \( \binom{6}{5} = 6 \)
- \( \binom{6}{6} = 1 \)

5. **Summing the Combinations**:
Now, we add all these values together:
\[ \text{Total Regions} = \binom{6}{1} + \binom{6}{2} + \binom{6}{3} + \binom{6}{4} + \binom{6}{5} + \binom{6}{6} \]
\[ = 6 + 15 + 20 + 15 + 6 + 1 = 63 \]

6. **Final Answer**:
The maximum possible number of regions created by 6 overlapping triangles is **63**.