PQRS is a quadrilateral having 3 4 5 6 points on PQ , QR, RS and SP respectively. The number of triangles with vertices on different sides is

Options

A. 220
B. 270
C. 282
D. none of these

Answer

To find the number of triangles that can be formed with vertices on different sides of the quadrilateral PQRS, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Points on Each Side:**
- Let’s denote the number of points on each side of the quadrilateral:
- Points on side PQ = 3
- Points on side QR = 4
- Points on side RS = 5
- Points on side SP = 6

2. **Determine the Combinations of Sides:**
- To form a triangle, we need to select 3 sides from the 4 available sides (PQ, QR, RS, SP). The combinations of sides can be calculated using the combination formula \( C(n, r) \), where \( n \) is the total number of sides and \( r \) is the number of sides to choose.
- The number of ways to choose 3 sides from 4 is:
\[ C(4, 3) = 4 \]

3. **Calculate the Number of Triangles for Each Combination of Sides:**
- We will calculate the number of triangles for each combination of sides by taking one point from each selected side.

- **Combination 1: PQ, QR, RS**
- Points = \( 3 \) (from PQ) × \( 4 \) (from QR) × \( 5 \) (from RS)
- Total triangles = \( 3 \times 4 \times 5 = 60 \)

- **Combination 2: QR, RS, SP**
- Points = \( 4 \) (from QR) × \( 5 \) (from RS) × \( 6 \) (from SP)
- Total triangles = \( 4 \times 5 \times 6 = 120 \)

- **Combination 3: RS, SP, PQ**
- Points = \( 5 \) (from RS) × \( 6 \) (from SP) × \( 3 \) (from PQ)
- Total triangles = \( 5 \times 6 \times 3 = 90 \)

- **Combination 4: SP, PQ, QR**
- Points = \( 6 \) (from SP) × \( 3 \) (from PQ) × \( 4 \) (from QR)
- Total triangles = \( 6 \times 3 \times 4 = 72 \)

4. **Sum the Total Number of Triangles:**
- Now, we will add the number of triangles from all combinations:
\[ \text{Total triangles} = 60 + 120 + 90 + 72 = 342 \]

5. **Conclusion:**
- The total number of triangles that can be formed with vertices on different sides of the quadrilateral PQRS is **342**.