The straight lines S1, S2, S3 are in a parallel and lie in the same plane. A total number of A points on S1, B points on S2 and C points on S3 are used to produce triangles. What is the maximum number of triangles formed?

Options

A. `""^(A+b+c)C_(3) -""^(A)C_(3) -""^(B)C_(3) - ""^(C )C_(3) +1`
B. `""^(A+b+c)C_(3)`
C. `""^(A+b+c)C_(3)+1`
D. `(""^(A+b+c)C_(3)-""^(A)C_(3)-""^(A)C_(3) -""^(B)C_(3) -""^(C )C_(3 ))`

Answer

To solve the problem, we need to determine the maximum number of triangles that can be formed using points on three parallel lines, S1, S2, and S3. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We have three parallel lines:
- Line S1 has A points.
- Line S2 has B points.
- Line S3 has C points.

Since the lines are parallel, any three points chosen from the same line will be collinear and cannot form a triangle.

### Step 2: Calculate Total Points
The total number of points available is:
\[ \text{Total Points} = A + B + C \]

### Step 3: Determine Combinations of Points
To form a triangle, we need to select 3 points from the total points. The number of ways to choose 3 points from \( A + B + C \) points is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Thus, the total combinations of points to form triangles is:
\[ \text{Total Combinations} = \binom{A + B + C}{3} \]

### Step 4: Subtract Collinear Combinations
However, we need to subtract the combinations that do not form triangles (i.e., combinations of points from the same line):
1. From line S1 (A points), the number of combinations is:
\[ \binom{A}{3} \]
2. From line S2 (B points), the number of combinations is:
\[ \binom{B}{3} \]
3. From line S3 (C points), the number of combinations is:
\[ \binom{C}{3} \]

### Step 5: Calculate the Actual Number of Triangles
The actual number of triangles that can be formed is given by subtracting the collinear combinations from the total combinations:
\[ \text{Number of Triangles} = \binom{A + B + C}{3} - \binom{A}{3} - \binom{B}{3} - \binom{C}{3} \]

### Final Answer
Thus, the maximum number of triangles that can be formed is:
\[ \text{Number of Triangles} = \binom{A + B + C}{3} - \binom{A}{3} - \binom{B}{3} - \binom{C}{3} \]