Let ABC be a triangle. If `P_1, P_2, P_3, P_4, P_5` are five points on side AB, `P_6, P_7,......P_(11)`, are 6 points on side BC and `P_(12), P_(13),.......P_(18)` , are 7 points on side AC then find the number of triangle formed by

Answer

To find the number of triangles formed by the points \( P_1, P_2, \ldots, P_{18} \) on the sides of triangle \( ABC \), we can follow these steps:

### Step 1: Count the Total Points
We have a total of 18 points:
- 5 points on side \( AB \) (i.e., \( P_1, P_2, P_3, P_4, P_5 \))
- 6 points on side \( BC \) (i.e., \( P_6, P_7, P_8, P_9, P_{10}, P_{11} \))
- 7 points on side \( AC \) (i.e., \( P_{12}, P_{13}, P_{14}, P_{15}, P_{16}, P_{17}, P_{18} \))

### Step 2: Calculate Total Combinations of Points
The total number of ways to choose 3 points from 18 points to form a triangle is given by the combination formula \( \binom{n}{r} \):
\[ \text{Total combinations} = \binom{18}{3} \]

Calculating \( \binom{18}{3} \):
\[ \binom{18}{3} = \frac{18 \times 17 \times 16}{3 \times 2 \times 1} = \frac{4896}{6} = 816 \]

### Step 3: Subtract Invalid Combinations
Triangles cannot be formed if all three points are collinear. We need to subtract the combinations of points that lie on the same side of the triangle.

1. **Points on side \( AB \)**:
- Number of ways to choose 3 points from 5 points:
\[ \binom{5}{3} = \frac{5 \times 4}{2 \times 1} = 10 \]

2. **Points on side \( BC \)**:
- Number of ways to choose 3 points from 6 points:
\[ \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

3. **Points on side \( AC \)**:
- Number of ways to choose 3 points from 7 points:
\[ \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

### Step 4: Calculate Total Invalid Combinations
Now, we sum the invalid combinations:
\[ \text{Total invalid combinations} = \binom{5}{3} + \binom{6}{3} + \binom{7}{3} = 10 + 20 + 35 = 65 \]

### Step 5: Calculate Valid Triangles
To find the number of valid triangles, we subtract the invalid combinations from the total combinations:
\[ \text{Valid triangles} = \binom{18}{3} - (\binom{5}{3} + \binom{6}{3} + \binom{7}{3}) = 816 - 65 = 751 \]

### Final Answer
The number of triangles formed by the points is \( \boxed{751} \).