Key Concepts and Formulas

• Combinations: The number of ways to choose $r$ objects from a set of $n$ distinct objects, without regard to order, is given by the combination formula: ${n \choose r} = \frac{n!}{r!(n-r)!}$.
• Triangle Formation: Three points form a triangle if and only if they are not collinear.

Step-by-Step Solution

Step 1: Calculate the total number of ways to choose 3 points from 22 points.

We have a total of $1 + 12 + 9 = 22$ points (origin + 12 points on $L_1$ + 9 points on $L_2$). We need to choose 3 points out of these 22 to form a triangle. The total number of ways to choose 3 points is: ${22 \choose 3} = \frac{22!}{3!19!} = \frac{22 \times 21 \times 20}{3 \times 2 \times 1} = 11 \times 7 \times 20 = 1540$ This represents all possible combinations of 3 points.

Step 2: Calculate the number of ways to choose 3 collinear points on L1.

If all three points are chosen from the 13 points on $L_1$ (the origin and the 12 points $P_i$), then they are collinear and do not form a triangle. The number of ways to choose 3 points from these 13 points is: ${13 \choose 3} = \frac{13!}{3!10!} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 13 \times 2 \times 11 = 286$

Step 3: Calculate the number of ways to choose 3 collinear points on L2.

Similarly, if all three points are chosen from the 10 points on $L_2$ (the origin and the 9 points $Q_i$), then they are collinear and do not form a triangle. The number of ways to choose 3 points from these 10 points is: ${10 \choose 3} = \frac{10!}{3!7!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 10 \times 3 \times 4 = 120$

Step 4: Calculate the number of triangles that can be formed.

To find the number of triangles, we subtract the number of collinear combinations from the total number of combinations: $\text{Number of triangles} = {22 \choose 3} - {13 \choose 3} - {10 \choose 3} = 1540 - 286 - 120 = 1540 - 406 = 1134$

Common Mistakes & Tips

• Collinear Points: Remember to subtract the cases where the chosen points are collinear, as they do not form triangles.
• Origin: Include the origin when considering collinear points on $L_1$ and $L_2$.
• Careful Calculation: Double-check your calculations, especially when dealing with combinations.

Summary

We calculated the total number of ways to choose 3 points from the 22 given points. Then, we subtracted the number of ways to choose 3 points that are collinear on line $L_1$ and line $L_2$. The resulting number represents the total number of triangles that can be formed. This gives us 1134 triangles.

Final Answer

The final answer is \boxed{1134}, which corresponds to option (C).