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 binomial coefficient: ${n \choose r} = \frac{n!}{r!(n-r)!}$
• Triangle Formation: Three non-collinear points are required to form a triangle.

Step-by-Step Solution

Step 1: Calculate the total number of ways to choose 3 points from the 18 points.

We have a total of 18 points. To form a triangle, we need to select 3 points. Therefore, the total number of ways to select any 3 points from these 18 points is given by ${18 \choose 3}$.

${18 \choose 3} = \frac{18!}{3!(18-3)!} = \frac{18!}{3!15!} = \frac{18 \times 17 \times 16}{3 \times 2 \times 1} = 3 \times 17 \times 16 = 816$

Step 2: Calculate the number of ways to choose 3 points that lie on side AB.

We have 5 points on side AB. Choosing any 3 of these points will not form a triangle because they are collinear. The number of ways to choose 3 points from these 5 is ${5 \choose 3}$.

${5 \choose 3} = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = \frac{5 \times 4}{2 \times 1} = 10$

Step 3: Calculate the number of ways to choose 3 points that lie on side BC.

We have 6 points on side BC. Choosing any 3 of these points will not form a triangle because they are collinear. The number of ways to choose 3 points from these 6 is ${6 \choose 3}$.

${6 \choose 3} = \frac{6!}{3!(6-3)!} = \frac{6!}{3!3!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$

Step 4: Calculate the number of ways to choose 3 points that lie on side CA.

We have 7 points on side CA. Choosing any 3 of these points will not form a triangle because they are collinear. The number of ways to choose 3 points from these 7 is ${7 \choose 3}$.

${7 \choose 3} = \frac{7!}{3!(7-3)!} = \frac{7!}{3!4!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$

Step 5: Calculate the number of triangles that can be formed.

To find the number of triangles that can be formed, we subtract the number of collinear point combinations from the total number of combinations of 3 points.

Number of triangles = Total combinations - (Combinations on AB) - (Combinations on BC) - (Combinations on CA)

$\text{Number of triangles} = {18 \choose 3} - {5 \choose 3} - {6 \choose 3} - {7 \choose 3} = 816 - 10 - 20 - 35 = 816 - 65 = 751$

Common Mistakes & Tips

• Collinear Points: Remember that three collinear points cannot form a triangle. You must subtract these cases from the total number of combinations.
• Combinations vs. Permutations: Since the order of selecting the points doesn't matter, use combinations (${n \choose r}$) instead of permutations.
• Careful Calculation: Double-check your calculations, especially when evaluating the binomial coefficients.

Summary

To find the number of triangles that can be formed using the given points, we first calculate the total number of ways to choose any three points from the set of 18 points. Then, we subtract the number of combinations where the three points are collinear (i.e., lie on the same side of the triangle). This gives us the number of valid triangles that can be formed. The total number of triangles is 751.

Final Answer

The final answer is $\boxed{751}$, which corresponds to option (A).