The number of circles that can be drawn out of 10 points of which 7 are collinear is

Options

A. 130
B. 85
C. 45
D. Cannot be determined

Answer

To solve the problem of finding the number of circles that can be drawn from 10 points, where 7 points are collinear, we can follow these steps:

### Step 1: Understand the requirement to form a circle
A unique circle can be formed by selecting any three non-collinear points. Since we have 7 collinear points, we cannot select all three points from these collinear points.

### Step 2: Identify the points
We have:
- 7 collinear points (let's call them A1, A2, A3, A4, A5, A6, A7)
- 3 non-collinear points (let's call them B1, B2, B3)

### Step 3: Calculate the different combinations to form circles
We can form circles in the following ways:

1. **Select 1 point from the collinear points and 2 points from the non-collinear points**:
- The number of ways to select 1 point from 7 collinear points: \( \binom{7}{1} = 7 \)
- The number of ways to select 2 points from 3 non-collinear points: \( \binom{3}{2} = 3 \)
- Total combinations for this case: \( 7 \times 3 = 21 \)

2. **Select 2 points from the collinear points and 1 point from the non-collinear points**:
- The number of ways to select 2 points from 7 collinear points: \( \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \)
- The number of ways to select 1 point from 3 non-collinear points: \( \binom{3}{1} = 3 \)
- Total combinations for this case: \( 21 \times 3 = 63 \)

3. **Select all 3 points from the non-collinear points**:
- The number of ways to select 3 points from 3 non-collinear points: \( \binom{3}{3} = 1 \)
- Total combinations for this case: \( 1 \)

### Step 4: Add all the combinations together
Now, we add all the combinations from the three cases:
- From case 1: 21
- From case 2: 63
- From case 3: 1

Total circles = \( 21 + 63 + 1 = 85 \)

### Conclusion
The total number of circles that can be formed is **85**.