To find the total number of triangles that can be formed using the given 13 points, we need to consider the different combinations of 3 points taken from these 13 points.

Given information:
- 6 points are on line X.
- 7 points are on line Y.
- Total number of points = 13.

Now, let's consider the different cases for forming triangles using these points.

Case 1: All 3 points lie on line X or line Y
- In this case, we cannot form a triangle as all the points are collinear.

Case 2: 2 points lie on line X and 1 point lies on line Y
- There are 6 points on line X and 7 points on line Y.
- We can choose 2 points from line X in 6C2 ways and choose 1 point from line Y in 7C1 ways.
- Total number of triangles = 6C2 * 7C1 = (6*5/2*1) * 7 = 15 * 7 = 105.

Case 3: 2 points lie on line Y and 1 point lies on line X
- Similar to Case 2, we can choose 2 points from line Y in 7C2 ways and choose 1 point from line X in 6C1 ways.
- Total number of triangles = 7C2 * 6C1 = (7*6/2*1) * 6 = 21 * 6 = 126.

Case 4: 1 point lies on line X, 1 point lies on line Y, and 1 point lies outside both lines
- There are 6 points on line X and 7 points on line Y.
- We can choose 1 point from line X in 6C1 ways, choose 1 point from line Y in 7C1 ways, and choose 1 point from the remaining 13-6-7=0 points in 0C1 way.
- Total number of triangles = 6C1 * 7C1 * 0C1 = 6 * 7 * 1 = 42.

Case 5: All 3 points lie outside both lines
- We can choose 3 points from the remaining 13-6-7=0 points in 0C3 ways.
- Total number of triangles = 0C3 = 0.

Hence, the total number of triangles that can be formed using the 13 points is:

Total triangles = Case 2 + Case 3 + Case 4 + Case 5 = 105 + 126 + 42 + 0 = 273.

Therefore, the correct answer is option B) 231.

⁷C₂ ​ ∗ 6 + ⁶C₂ ​∗ 7 = 231