In a plane there are 8 points , no three of which are collinear .How many lines do the points determine ?

Options

A. 7
B. 16
C. 28
D. 36

Explanation

To solve the problem of how many lines can be determined by 8 points in a plane where no three points are collinear, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have 8 points in a plane, and we know that no three points are collinear. This means that any two points will form a unique line.

2. **Choosing Points to Form Lines**:
A line is determined by any two points. Therefore, we need to find out how many ways we can choose 2 points from the 8 points.

3. **Using Combinations**:
The number of ways to choose 2 points from 8 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of points, and \( r \) is the number of points to choose.

4. **Applying the Formula**:
Here, \( n = 8 \) and \( r = 2 \):
\[ \binom{8}{2} = \frac{8!}{2!(8-2)!} = \frac{8!}{2! \cdot 6!} \]

5. **Simplifying the Factorials**:
We can simplify the factorials:
\[ \binom{8}{2} = \frac{8 \times 7 \times 6!}{2! \times 6!} \]
The \( 6! \) cancels out:
\[ = \frac{8 \times 7}{2!} \]

6. **Calculating \( 2! \)**:
We know that \( 2! = 2 \):
\[ = \frac{8 \times 7}{2} = \frac{56}{2} = 28 \]

7. **Conclusion**:
Therefore, the total number of lines determined by the 8 points is \( 28 \).

### Final Answer:
The number of lines determined by the 8 points is **28**.

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