A triangle ABC in which side AB,BC,CA consist 5,3,6 points respectively, then the number of triangle that can be formed by these points are

Options

A. 360
B. 333
C. 396
D. 320

Answer

To find the number of triangles that can be formed using the points on the sides of triangle ABC, we will follow these steps:

### Step 1: Identify the Points
We have the following points on each side of triangle ABC:
- Side AB has 5 points.
- Side BC has 3 points.
- Side CA has 6 points.

### Step 2: Calculate Total Points
The total number of points available is:
\[ \text{Total Points} = 5 + 3 + 6 = 14 \]

### Step 3: Calculate Total Combinations of Points
To form a triangle, we need to select 3 points from these 14 points. The number of ways to choose 3 points from 14 is given by the combination formula \( \binom{n}{r} \):
\[ \text{Total Combinations} = \binom{14}{3} \]

Calculating \( \binom{14}{3} \):
\[ \binom{14}{3} = \frac{14 \times 13 \times 12}{3 \times 2 \times 1} = \frac{2184}{6} = 364 \]

### Step 4: Subtract Invalid Combinations
However, we need to subtract the cases where all 3 points are chosen from the same side, as they do not form a triangle.

1. **Points from Side AB (5 points)**:
\[ \text{Invalid from AB} = \binom{5}{3} = \frac{5 \times 4}{2 \times 1} = 10 \]

2. **Points from Side BC (3 points)**:
\[ \text{Invalid from BC} = \binom{3}{3} = 1 \]

3. **Points from Side CA (6 points)**:
\[ \text{Invalid from CA} = \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

### Step 5: Total Invalid Combinations
Now, we sum the invalid combinations:
\[ \text{Total Invalid} = 10 + 1 + 20 = 31 \]

### Step 6: Calculate Valid Combinations
Finally, we subtract the invalid combinations from the total combinations:
\[ \text{Valid Combinations} = \text{Total Combinations} - \text{Total Invalid} = 364 - 31 = 333 \]

### Conclusion
The total number of triangles that can be formed using these points is:
\[ \boxed{333} \]