The sides AB, BC, CA, of a triangle ABC have 3, 4 and 5 interior points respectively on them. The total number of triangles that can be constructed by using these points as vertices is

Options

A. A. 105
B. B. 205
C. C. 221
D. D. none of these

Answer

To solve the problem of finding the total number of triangles that can be constructed using the points on the sides of triangle ABC, we can follow these steps:

### Step 1: Identify the Points
We have the following points on each side of triangle ABC:
- Side AB has 3 interior points.
- Side BC has 4 interior points.
- Side CA has 5 interior points.

### Step 2: Calculate the Total Number of Points
The total number of points on the triangle is the sum of the points on each side:
\[ \text{Total Points} = 3 + 4 + 5 = 12 \]

### Step 3: Calculate the Total Combinations of Points
To form a triangle, we need to choose 3 points from the total of 12 points. The number of ways to choose 3 points from 12 is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Where \( n \) is the total number of points and \( r \) is the number of points to choose. Thus, we calculate:
\[ \binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \]

### Step 4: Subtract Collinear Combinations
Next, we need to subtract the combinations that do not form triangles because they are collinear:
- For side AB (3 points), the number of combinations is:
\[ \binom{3}{3} = 1 \text{ (only one triangle can be formed with all points on AB)} \]
- For side BC (4 points), the number of combinations is:
\[ \binom{4}{3} = 4 \text{ (four triangles can be formed with all points on BC)} \]
- For side CA (5 points), the number of combinations is:
\[ \binom{5}{3} = 10 \text{ (ten triangles can be formed with all points on CA)} \]

### Step 5: Total Collinear Combinations
Now, we sum the collinear combinations:
\[ \text{Total Collinear Combinations} = 1 + 4 + 10 = 15 \]

### Step 6: Calculate Valid Triangles
Finally, we subtract the collinear combinations from the total combinations to find the valid triangles:
\[ \text{Valid Triangles} = 220 - 15 = 205 \]

### Final Answer
The total number of triangles that can be constructed using these points as vertices is **205**.

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