Key Concepts and Formulas

• Combinations: The number of ways to choose $k$ objects from a set of $n$ distinct objects, without regard to order, is given by $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
• Triangle Formation: Three non-collinear points uniquely define a triangle.
• Subtraction Principle: To find the number of ways to do something with restrictions, calculate the total number of ways without restrictions, and then subtract the number of ways that violate the restrictions.

Step-by-Step Solution

Step 1: Calculate the total number of ways to choose 3 points from 12 without any restrictions.

• Why this step? We first find the total number of possible combinations of 3 points that can be selected from the 12 given points, before considering the collinearity restriction. This will be our starting point.
• Using the combination formula with $n = 12$ and $k = 3$: $\binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12!}{3!9!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 2 \times 11 \times 10 = 220$
• This means that there are 220 ways to select any 3 points from the 12 points.

Step 2: Calculate the number of ways to choose 3 points from the 5 collinear points.

• Why this step? Since 5 points are collinear, any selection of 3 points from these 5 will not form a triangle. We need to calculate the number of these invalid combinations to subtract them from the total calculated in Step 1.
• Using the combination formula with $n = 5$ and $k = 3$: $\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = \frac{5 \times 4}{2 \times 1} = 10$
• There are 10 ways to choose 3 points from the 5 collinear points, which do not form triangles.

Step 3: Calculate the number of triangles that can be formed.

• Why this step? To obtain the number of valid triangles, we subtract the number of invalid combinations (3 collinear points) from the total number of combinations of 3 points.
• Number of triangles = (Total number of ways to choose 3 points) - (Number of ways to choose 3 collinear points) $\text{Number of Triangles} = 220 - 10 = 210$
• Therefore, the total number of triangles that can be formed is 210.

Common Mistakes & Tips

• Forgetting the Collinearity Condition: The most common mistake is to calculate $\binom{12}{3}$ and forget to account for the collinear points. Always pay close attention to such restrictions.
• Misunderstanding Combinations: Ensure you are using combinations (order doesn't matter) and not permutations (order matters). The problem asks for the number of triangles, and the order in which the vertices are chosen doesn't change the triangle.
• Double-Checking Calculations: Carefully re-calculate the combinations to avoid arithmetic errors.

Summary

The problem involves finding the number of triangles that can be formed from 12 points, with the restriction that 5 of the points are collinear. We first calculate the total number of ways to choose 3 points without considering the restriction, and then subtract the number of ways to choose 3 points from the collinear set, as these do not form triangles. This gives us the final answer of 210.

The final answer is $\boxed{210}$, which corresponds to option (B).