Let `L_1 and L_2` be two lines intersecting at P If `A_1,B_1,C_1` are points on `L_1,A_2, B_2,C_2,D_2,E_2` are points on `L_2` and if none of these coincides with P, then the number of triangles formed by these 8 points is

Options

A. 56
B. 55
C. 46
D. 45

Answer

To solve the problem of finding the number of triangles formed by the points on two intersecting lines, we can follow these steps:

### Step 1: Understand the total number of points
We have a total of 8 points:
- 3 points on line L1: \( A_1, B_1, C_1 \)
- 5 points on line L2: \( A_2, B_2, C_2, D_2, E_2 \)

### Step 2: Calculate the total number of triangles from 8 points
The total number of ways to choose 3 points from 8 points to form a triangle 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.

\[ \text{Total triangles} = \binom{8}{3} \]

Calculating \( \binom{8}{3} \):

\[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]

### Step 3: Subtract triangles formed by points on line L1
Next, we need to find the number of triangles that can be formed using only the points on line L1. Since there are only 3 points on line L1, the number of triangles that can be formed is:

\[ \text{Triangles from } L1 = \binom{3}{3} \]

Calculating \( \binom{3}{3} \):

\[ \binom{3}{3} = 1 \]

### Step 4: Subtract triangles formed by points on line L2
Now, we calculate the number of triangles that can be formed using only the points on line L2. There are 5 points on line L2, so the number of triangles that can be formed is:

\[ \text{Triangles from } L2 = \binom{5}{3} \]

Calculating \( \binom{5}{3} \):

\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 5: Calculate the final number of triangles
Now, we can find the total number of triangles that can be formed by subtracting the triangles formed by points on L1 and L2 from the total number of triangles:

\[ \text{Total valid triangles} = \text{Total triangles} - \text{Triangles from } L1 - \text{Triangles from } L2 \]

Substituting the values we calculated:

\[ \text{Total valid triangles} = 56 - 1 - 10 = 45 \]

### Final Answer
Thus, the number of triangles formed by the 8 points is **45**.

---