4 points out of 8 points in a plane are collinear. Number of different quadrilateral that can be formed by joining them is

Options

A. 56
B. 60
C. 76
D. 53

Answer

To solve the problem of finding the number of different quadrilaterals that can be formed by joining 8 points in a plane, where 4 of the points are collinear, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Total Points and Collinear Points**:
- We have a total of 8 points, out of which 4 points are collinear. Let's denote the collinear points as A, B, C, and D, and the non-collinear points as E, F, G, and H.

2. **Calculate Total Combinations of 4 Points**:
- The total number of ways to choose 4 points from 8 points is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of points and \( r \) is the number of points to choose.
- Thus, the total combinations are:
\[ \binom{8}{4} = \frac{8!}{4! \cdot (8-4)!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]

3. **Subtract Invalid Combinations**:
- **Case 1**: All 4 points are collinear (A, B, C, D). These do not form a quadrilateral, so we need to subtract this case:
\[ \binom{4}{4} = 1 \]
- **Case 2**: Choosing 3 collinear points and 1 non-collinear point. This also does not form a quadrilateral. The number of ways to choose 3 collinear points from 4 and 1 from the 4 non-collinear points is:
\[ \binom{4}{3} \cdot \binom{4}{1} = 4 \cdot 4 = 16 \]

4. **Calculate the Valid Combinations**:
- Now we can calculate the total valid combinations of quadrilaterals:
\[ \text{Valid combinations} = \binom{8}{4} - \binom{4}{4} - \left(\binom{4}{3} \cdot \binom{4}{1}\right) \]
\[ = 70 - 1 - 16 = 53 \]

### Final Answer:
The number of different quadrilaterals that can be formed by joining the points is **53**.