There are 10 points on a line and 11 points on another line, which are parallel to each other. How many triangles can be drawn taking the vertices on any of the line?

Options

A. 1050
B. 2550
C. 150
D. 1045

Answer

To solve the problem of how many triangles can be formed using the points on two parallel lines, we need to consider the combinations of points selected from each line.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two lines: one with 10 points and another with 11 points. To form a triangle, we need to select 3 points. However, we cannot select all 3 points from the same line because they would be collinear (and thus not form a triangle). Therefore, we can either select:
- 3 points from the first line (not allowed, as they are collinear).
- 3 points from the second line (not allowed, as they are collinear).
- 2 points from the first line and 1 point from the second line.
- 1 point from the first line and 2 points from the second line.

2. **Calculating the Combinations**:
- **Case 1**: Selecting 2 points from the first line (10 points) and 1 point from the second line (11 points).
\[ \text{Number of ways} = \binom{10}{2} \times \binom{11}{1} \]
\[ = \frac{10 \times 9}{2 \times 1} \times 11 = 45 \times 11 = 495 \]

- **Case 2**: Selecting 1 point from the first line (10 points) and 2 points from the second line (11 points).
\[ \text{Number of ways} = \binom{10}{1} \times \binom{11}{2} \]
\[ = 10 \times \frac{11 \times 10}{2 \times 1} = 10 \times 55 = 550 \]

3. **Total Number of Triangles**:
Now, we add the results from both cases to find the total number of triangles that can be formed:
\[ \text{Total triangles} = 495 + 550 = 1045 \]

### Final Answer:
The total number of triangles that can be drawn using the vertices on the two parallel lines is **1045**.