There are 10 points in a plane, no three are collinear, except 4 which are collinear. All points are joined. Let L be the number of different straight lines and T be the number of different triangles, then

Options

A. A. T=120
B. B. L=40
C. C. T=3L−5
D. D. data insufficient

Answer

To solve the problem, we need to find the number of different straight lines (L) and the number of different triangles (T) that can be formed with the given points.

### Step 1: Calculate the number of straight lines (L)

1. **Understanding the points**: We have a total of 10 points, out of which 4 points are collinear. The remaining 6 points are not collinear with each other or with the 4 collinear points.

2. **Finding lines from non-collinear points**:
- We can choose any 2 points from the 6 non-collinear points to form a line. The number of ways to choose 2 points from 6 is given by the combination formula \( \binom{n}{r} \):
\[ \text{Lines from non-collinear points} = \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \]

3. **Finding lines from collinear points**:
- The 4 collinear points form 1 line.

4. **Total lines (L)**:
- Therefore, the total number of different straight lines is:
\[ L = 15 + 1 = 16 \]

### Step 2: Calculate the number of triangles (T)

1. **Understanding triangle formation**: A triangle is formed by selecting any 3 points. However, we cannot form a triangle using all 4 collinear points since they lie on the same line.

2. **Finding triangles from all points**:
- The total number of triangles that can be formed from 10 points is:
\[ \text{Total triangles} = \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]

3. **Subtracting invalid triangles**:
- The only invalid triangles are those formed by the 4 collinear points. The number of ways to choose 3 points from these 4 collinear points is:
\[ \text{Invalid triangles} = \binom{4}{3} = 4 \]

4. **Total triangles (T)**:
- Therefore, the total number of different triangles is:
\[ T = 120 - 4 = 116 \]

### Final Results

- The number of different straight lines \( L = 16 \)
- The number of different triangles \( T = 116 \)