The straight line $l_{1}$, $l_{2}$, $l_{3}$ are parallel and lie in the same plane. A total number of m points are taken on $l_{1}$, n points on $l_{2}$, k points on $l_{3}$. The maximum number of triangles formed with vertices at these points are:

Question

Accepted Answer

To determine the maximum number of triangles that can be formed with vertices at the given points on the parallel lines $ l_1 $, $ l_2 $, and $ l_3 $, we need to consider the conditions under which a triangle can be formed. A triangle requires three non-collinear points, meaning the points must not all lie on the same line.

### Explanation for Option 3:

**Option 3: ${}^{m+n+k}C_{3} - {}^{m}C_{3} - {}^{n}C_{3} - {}^{k}C_{3}$**

1. **Total Combinations of Points:** The expression ${}^{m+n+k}C_{3}$ represents the total number of ways to choose any three points from the combined set of points on all three lines. This includes all possible combinations, regardless of whether the points are collinear or not.

2. **Subtracting Collinear Combinations:**
   - ${}^{m}C_{3}$ represents the number of ways to choose three points from line $ l_1 $. Any such combination is collinear and cannot form a triangle.
   - ${}^{n}C_{3}$ represents the number of ways to choose three points from line $ l_2 $. These are also collinear.
   - ${}^{k}C_{3}$ represents the number of ways to choose three points from line $ l_3 $. These are collinear as well.

3. **Correct Calculation of Non-Collinear Triangles:** By subtracting the collinear combinations from the total combinations, we are left with only those combinations that can form triangles. Thus, the expression ${}^{m+n+k}C_{3} - {}^{m}C_{3} - {}^{n}C_{3} - {}^{k}C_{3}$ correctly calculates the maximum number of triangles that can be formed with vertices at these points.

### Explanation for Why Other Options are Incorrect:

**Option 1: ${}^{m}C_{3} + {}^{n}C_{3} + {}^{k}C_{3}$**

- This option only counts the number of ways to choose three points from each individual line, which are all collinear. Therefore, it does not account for any triangles, as no triangles can be formed from three collinear points.

**Option 2: ${}^{m+n+k}C_{3}$**

- This option counts all possible combinations of three points from the total set of points, including those that are collinear. It does not exclude the collinear combinations, which means it overestimates the number of triangles.

**Option 4: ${}^{m+n+k}C_{3} + {}^{m}C_{3} + {}^{n}C_{3} + {}^{k}C_{3}$**

- This option adds the collinear combinations to the total combinations, further overestimating the number of triangles. It includes both non-collinear and collinear combinations, which is incorrect for counting triangles.

In conclusion, Option 3 is the correct answer because it accurately accounts for the non-collinear combinations of points, which are necessary to form triangles, by subtracting the collinear combinations from the total possible combinations.