Key Concepts and Formulas

• Combinations: The number of ways to choose $r$ objects from a set of $n$ objects without regard to order is given by the combination formula: ${}^{n}{C_r} = \frac{n!}{r!(n-r)!}$
• Triangle Formation: A triangle is formed by selecting three non-collinear points.
• Inclusion-Exclusion Principle: To count the number of objects satisfying a condition, subtract the number of objects that do not satisfy the condition from the total number of objects.

Step-by-Step Solution

Step 1: Calculate the total number of ways to select 3 points from 10.

We need to find the total number of ways to choose 3 points out of 10, irrespective of whether they form a triangle or not. This is a combination problem since the order of selection does not matter.

${}^{10}{C_3} = \frac{10!}{3!(10-3)!} = \frac{10!}{3!7!}$ ${}^{10}{C_3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 10 \times 3 \times 4 = 120$ Thus, there are 120 ways to choose 3 points from 10 points.

Step 2: Calculate the number of ways to select 3 collinear points from the 6 collinear points.

Since 6 points are collinear, any selection of 3 points from these 6 will not form a triangle. We need to subtract these invalid selections from the total calculated in Step 1.

${}^{6}{C_3} = \frac{6!}{3!(6-3)!} = \frac{6!}{3!3!}$ ${}^{6}{C_3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$ There are 20 ways to choose 3 points from the 6 collinear points.

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

To find the number of triangles, we subtract the number of invalid selections (3 collinear points) from the total number of selections (3 points from 10).

$N = {}^{10}{C_3} - {}^{6}{C_3} = 120 - 20 = 100$ Therefore, the number of triangles that can be formed is 100.

Step 4: Compare with Given Options

We have found that $N=100$. Now, we compare this value with the options provided:

(A) $N \le 100$ (B) $100 < N \le 140$ (C) $140 < N \le 190$ (D) $N > 190$

Since $100 \le 100$, option (A) is the correct choice.

Common Mistakes & Tips

• Forgetting to subtract: A common mistake is to calculate only ${}^{10}C_3$ and forget to subtract the combinations of collinear points.
• Combinations vs. Permutations: Always use combinations when the order of selection doesn't matter.
• Understanding Collinearity: Always remember that three collinear points do not form a triangle.

Summary

To find the number of triangles formed by 10 points, where 6 are collinear, we first calculated the total number of ways to choose 3 points from the 10 points (${}^{10}C_3 = 120$). Then, we subtracted the number of ways to choose 3 points from the 6 collinear points (${}^{6}C_3 = 20$), as these selections do not form triangles. This gave us the number of valid triangles, $N = 120 - 20 = 100$. Comparing this result with the given options, we found that option (A) is correct.

Final Answer

The final answer is \boxed{100}, which corresponds to option (A).