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NDA Maths · Permutation & Combination

Geometric Counting

Counting lines, triangles, quadrilaterals, diagonals, and intersection points from a set of points or lines — combinations with a correction for collinear (degenerate) cases.

All Permutation & Combination notesNDA Maths strategy

Why this matters

Geometric counting is pure combination with one twist: collinear points make no triangle and concurrent/parallel lines lose intersections. Subtract the degenerate cases and these become routine.

Concept 1 of 1

Lines, triangles and polygons from points

Intuition

A line needs 2 points, a triangle 3, a quadrilateral 4 — so the raw counts are nC2, nC3, nC4^nC_2,\,^nC_3,\,^nC_4. The correction: any set of collinear points that 'should' form a figure doesn't, so subtract those degenerate selections.

Definition

From nn points, no three collinear: lines nC2^nC_2, triangles nC3^nC_3, quadrilaterals nC4^nC_4. **If kk points are collinear:** subtract their degenerate selections — lines nC2−kC2+1^nC_2-^kC_2+1; triangles nC3−kC3^nC_3-^kC_3. Diagonals of an nn-gon: nC2−n=n(n−3)2^nC_2-n=\dfrac{n(n-3)}{2}. Parallelograms from mm and nn parallel lines: mC2⋅nC2^mC_2\cdot{}^nC_2. Max intersection points of nn lines: nC2^nC_2.

triangles = ⁸C₃ − ³C₃ (drop the collinear triple)

Worked example

How many triangles can be formed from 8 points, of which 3 are collinear?
  1. All triples: 8C3=56^8C_3=56; degenerate (the 3 collinear): 3C3=1^3C_3=1.
  2. 56−1=5556-1=55.
Answer:5555.
Practice this conceptself-check · 4 quick reps

Try it yourself

How many triangles from 12 points, of which 7 are collinear?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

  1. 1.
    Triangles from nn points (no 3 collinear)?
  2. 2.
    Correction for kk collinear points (triangles)?
  3. 3.
    Diagonals of an nn-gon?
  4. 4.
    Parallelograms from mm and nn parallel lines?

From the bank · past-year question

Example 1Permutation & CombinationMODERATE
For the following two (02) items: There are 8 points on a plane out of which 4 points are collinear.
How many triangles can be formed by joining these points?

[Q35 · Sep · 2025]

Drill 12 more on lines, triangles and polygons from points

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Permutation & CombinationEASY
Suppose 20 distinct points are placed randomly on a circle. Which of the following statements is/are correct? 1. The number of straight lines that can be drawn by joining any two of these points is 380. 2. The number of triangles that can be drawn by joining any three of these points is 1140. Select the correct answer using the code given below.

[Q19 · Sep · 2021]

Example 2Permutation & CombinationEASY
What is the maximum number of points of intersection of 10 circles?

[Q13 · Sep · 2023]

Example 3Permutation & CombinationMODERATE
For the following two (02) items: There are 8 points on a plane out of which 4 points are collinear.
How many quadrilaterals can be formed by joining these points?

[Q36 · Sep · 2025]

Example 4Permutation & CombinationEASY
What is the number of diagonals of an octagon ?

[Q10 · Sep · 2019]

Example 5Permutation & CombinationHARD
Consider a regular polygon with 10 sides. What is the number of triangles that can be formed by joining the vertices which have no common side with any of the sides of the polygon?

[Q25 · Sep · 2021]

Drill every past-year question on this subtopic

13 questions from the bank — paginated, with cart and Word-export support.

Drill the 13 questions

Related notes