JEE Mains Maths · Permutations and Combinations

Points, Lines and Polygons

Counting triangles, quadrilaterals and intersection points formed by given points or lines, taking care of points that lie on one line and of the sides of a polygon.

All Permutations and Combinations notes JEE Mains Maths strategy

Why this matters

Ten PYQs, eight of them multiple choice. Every one is a choice of vertices minus the choices that fail — three collinear points make no triangle, parallel lines never meet, concurrent lines meet at one point. Six involve collinear points or special lines; four use the vertices of a polygon. Two ideas cover the page.

Concept 1 of 2: Collinear points, parallel and concurrent lines

Triangles from

n

points are

\binom n3

minus the collinear triples: subtract

\binom k3

for each line holding

k

points. Alternatively, when points lie on the sides of a figure, choose how many come from each side (at most two from any one side for a triangle's vertex set to work). For lines,

\binom n2

pairs meet unless parallel;

k

concurrent lines give 1 point instead of

\binom k2

Definition

Triangles: $\binom n3-\sum\binom{k_i}3$ .
Vertices taken from points on the sides of a figure: at most two from any one side, since three would be collinear.
Intersections of $n$ lines: $\binom n2$ ; $k$ parallel lose $\binom k2$ ; $k$ concurrent lose $\binom k2-1$ .

Triangles from points with collinear sets

\binom n3-\sum_i\binom{k_i}{3}

Worked example

Points: 4 on one line and 3 on another line, with no common point. How many triangles?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2021 · Paper 8 · Q71Moderate

Example 1 · Permutations and Combinations · Points, Lines and Polygons

If the sides

AB,BC

and

CA

of a triangle

ABC

have 3,5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to:

Vertices are not among the points

When the points lie in the interior of the sides, the triangle's own vertices are not available. Use only the given points, and subtract collinear triples side by side.

Drill 5 more on collinear points, parallel and concurrent lines

Concept 2 of 2: Triangles and diagonals in a polygon

Any 3 vertices of an

n

-gon form a triangle, so there are

\binom n3

of them; any 4 give a quadrilateral. Diagonals are

\binom n2-n

. Triangles using no side of the polygon are

\frac{n(n-4)(n-5)}{6}

; with exactly one side,

n(n-4)

; with two sides,

n

Definition

Triangles: $\binom n3$ . Quadrilaterals: $\binom n4$ .
Diagonals: $\binom n2-n=\frac{n(n-3)}{2}$ .
Triangles with no side of the polygon: $\frac{n(n-4)(n-5)}{6}$ .
$\binom{n+1}3-\binom n3=\binom n2$ ; $\binom n3+\binom n4=\binom{n+1}4$ .

No side of the polygon

\frac{n(n-4)(n-5)}{6}

Worked example

How many diagonals does a decagon have?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 6 April 2024 · Q76Moderate

Example 2 · Permutations and Combinations · Points, Lines and Polygons

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

Indices on a circle

For points

P_1,\dots,P_n

on a circle, any three give a triangle; a condition on the indices (like

i+j+k\neq15

) removes only the listed triples. List them carefully, with distinct indices.

Drill 3 more on triangles and diagonals in a polygon

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Collinear points, parallel and concurrent lines
Triangles from points with collinear sets
$\binom n3-\sum_i\binom{k_i}{3}$
Triangles and diagonals in a polygon
No side of the polygon
$\frac{n(n-4)(n-5)}{6}$

Watch out for (2)

Vertices are not among the points
→ Collinear points, parallel and concurrent lines
Indices on a circle
→ Triangles and diagonals in a polygon

Test yourself on Permutations and Combinations

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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