The sides of a triangle have 4, 5 and 6 interior points marked on them respectively. The total number of triangles that can be formed using any of these points

Options

A. 371
B. 415
C. 286
D. 421

Answer

To solve the problem of finding the total number of triangles that can be formed using the interior points marked on the sides of a triangle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Points on Each Side:**
- The triangle has 4 interior points on one side, 5 on another side, and 6 on the third side.
- Total points = 4 + 5 + 6 = 15 points.

2. **Determine the Total Combinations of Points:**
- To form a triangle, we need to select 3 points from the total of 15 points.
- The number of ways to choose 3 points from 15 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of points and \( r \) is the number of points to choose.
- Therefore, we need to calculate \( \binom{15}{3} \).

3. **Calculate \( \binom{15}{3} \):**
- The formula for combinations is:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
- Substituting the values:
\[ \binom{15}{3} = \frac{15!}{3!(15-3)!} = \frac{15!}{3! \cdot 12!} \]
- This simplifies to:
\[ \binom{15}{3} = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} = \frac{2730}{6} = 455 \]

4. **Identify Collinear Points:**
- We need to subtract the combinations of collinear points, which cannot form a triangle.
- For the side with 4 points, the number of ways to choose 3 points is \( \binom{4}{3} = 4 \).
- For the side with 5 points, the number of ways to choose 3 points is \( \binom{5}{3} = 10 \).
- For the side with 6 points, the number of ways to choose 3 points is \( \binom{6}{3} = 20 \).

5. **Calculate Total Collinear Combinations:**
- Total collinear combinations = \( 4 + 10 + 20 = 34 \).

6. **Calculate the Total Valid Triangles:**
- The total number of triangles that can be formed is:
\[ \text{Total triangles} = \text{Total combinations} - \text{Collinear combinations} = 455 - 34 = 421 \]

### Final Answer:
The total number of triangles that can be formed using the interior points is **421**.