Geometrical Combinatorics: Lines, Triangles, and Polygons
Master the core concepts of selection minus degenerate cases with formulas, examples, and JEE-level practice problems.
Why Geometrical Combinatorics Matters for JEE
Geometrical combinatorics bridges combinatorial mathematics with geometry, appearing frequently in both JEE Mains and Advanced. These problems test your understanding of:
- Selection principles applied to geometric configurations
- Elimination of degenerate cases (collinear points, etc.)
- Systematic counting approaches for complex geometric figures
- Formula application with careful condition checking
Number of Straight Lines from n Points
Finding maximum and actual number of straight lines formed by n points in a plane.
Key Formula:
Number of lines = $^nC_2 - ^mC_2 + 1$
Where m points are collinear (lie on the same straight line)
Understanding the Formula:
Maximum lines from n points = $^nC_2$ (if no three points are collinear)
When m points are collinear:
- We lose $^mC_2 - 1$ lines (all combinations of m points give only 1 line instead of mC₂ lines)
- Add 1 for the single line through all collinear points
Example 1: 10 points, no three collinear
Step 1: Maximum lines = $^{10}C_2 = \frac{10 \times 9}{2} = 45$
Step 2: Since no three collinear, all 45 lines are distinct
Answer: 45 straight lines
Example 2: 8 points with 5 collinear
Step 1: Maximum lines if no collinear = $^8C_2 = 28$
Step 2: Apply formula: Lines = $^8C_2 - ^5C_2 + 1$
Step 3: Calculate: $28 - 10 + 1 = 19$
Verification: From 5 collinear points, we get only 1 line instead of 10 possible lines
Answer: 19 straight lines
Number of Triangles from n Points
Finding triangles formed by n points with various collinear conditions.
Key Formula:
Number of triangles = $^nC_3 - ^mC_3$
Where m points are collinear (cannot form triangles)
Understanding the Formula:
Maximum triangles from n points = $^nC_3$ (if no three points are collinear)
When m points are collinear:
- We subtract $^mC_3$ because any 3 points from m collinear points cannot form a triangle
- The formula eliminates these "degenerate" cases
Example 1: 12 points, no three collinear
Step 1: Maximum triangles = $^{12}C_3 = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220$
Answer: 220 triangles
Example 2: 10 points with 4 collinear
Step 1: Maximum triangles = $^{10}C_3 = 120$
Step 2: Apply formula: Triangles = $^{10}C_3 - ^4C_3$
Step 3: Calculate: $120 - 4 = 116$
Verification: From 4 collinear points, any 3 points give 0 triangles (4C₃ = 4 invalid combinations)
Answer: 116 triangles
Example 3: 15 points with two sets: 5 collinear and 4 collinear
Step 1: Maximum triangles = $^{15}C_3 = 455$
Step 2: Subtract invalid triangles from both collinear sets
Step 3: Triangles = $^{15}C_3 - ^5C_3 - ^4C_3$
Step 4: Calculate: $455 - 10 - 4 = 441$
Answer: 441 triangles
Number of Diagonals in an n-sided Polygon
Finding diagonals in convex polygons and special cases.
Key Formula:
Number of diagonals = $^nC_2 - n = \frac{n(n-3)}{2}$
Total lines minus sides of the polygon
Understanding the Formula:
From n vertices, total lines = $^nC_2$
We subtract n because:
- n of these lines are sides of the polygon
- The remaining are diagonals
Example 1: Hexagon (6-sided polygon)
Step 1: Using formula: Diagonals = $\frac{6(6-3)}{2} = \frac{6 \times 3}{2} = 9$
Step 2: Verification: Total lines = $^6C_2 = 15$, minus 6 sides = 9 diagonals
Answer: 9 diagonals
Example 2: Decagon (10-sided polygon)
Step 1: Using formula: Diagonals = $\frac{10(10-3)}{2} = \frac{10 \times 7}{2} = 35$
Answer: 35 diagonals
Example 3: Number of diagonals that can be drawn from one vertex
Step 1: From one vertex, we can draw diagonals to all other vertices except:
• The vertex itself
• The two adjacent vertices (these would be sides)
Step 2: Diagonals from one vertex = $n - 3$
Example: For octagon (n=8): $8 - 3 = 5$ diagonals from one vertex
🚀 Problem-Solving Strategies
For Lines & Triangles:
- Always check for collinear points first
- Remember: 2 points always give 1 line
- 3 collinear points give 0 triangles
- For multiple collinear sets, subtract each separately
For Polygons:
- Diagonals = Total lines − Sides
- From one vertex: n − 3 diagonals
- Regular polygons have equal diagonal lengths
- Convex polygons have all diagonals inside
Advanced Applications Available
Includes intersection points of diagonals, triangles in polygons, and special geometrical configurations
📝 Quick Self-Test
Try these JEE-level problems to test your understanding:
1. Find number of lines from 12 points with exactly 4 collinear
2. How many triangles from 15 points with 6 collinear?
3. Find diagonals in a 12-sided polygon and from one vertex
4. 20 points with two collinear sets: 5 and 4 points. Find triangles.
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