Geometry Based Permutation and Combination with Examples
Geometry Based Permutation and Combination applies selection and arrangement formulas to points, lines, polygons, diagonals, chords and geometric figures. The main methods use combinations because geometric objects are usually selected without order. Special care is needed for collinear points, repeated intersections and circular arrangements.
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What is Geometry Based Permutation and Combination?
For example, any three non-collinear points determine one triangle, so n points with no three collinear points form C(n, 3) triangles. Similarly, joining every pair of n points gives C(n, 2) line segments or chords. If some points are collinear, selections that do not form the required figure must be subtracted.
Geometry Based Permutation and Combination Video Lessons
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Geometry-Based Permutation and Combination
Learn how to solve permutation and combination problems involving geometric arrangements, using appropriate counting methods to determine possible configurations.
Geometry Based Permutation and Combination Formula & Tricks
Important Formulas
Use this when r points, vertices or objects are selected from n objects and their order does not matter.
This applies when all n points are vertices and no three selected points are collinear.
Every pair of vertices gives a segment, but the n sides are not diagonals.
Each segment is determined by choosing two distinct endpoints.
This maximum applies when no three chords meet at an interior point and every pair of chords with distinct endpoints intersects inside the circle.
Quick Tricks
A triangle is determined by selecting three vertices. Do not arrange the three vertices, because different orders represent the same triangle.
First count all pairs of polygon vertices using C(n, 2). Remove the n adjacent pairs that form sides.
If k points lie on one straight line, C(k, 3) selections from that line do not form triangles. Subtract these invalid selections from the total.
The expression 1 + C(n, 2) + C(n, 4) assumes that no three chords are concurrent inside the circle. If three or more chords meet at one interior point, the actual number of regions is smaller.
Geometry Based Permutation and Combination Concepts
Triangles from Points or Polygon Vertices
The selection is a combination because the order of the three chosen points does not matter. If points lie on the same straight line, their selection is invalid for triangle formation and must be removed.
Effect of Collinear Points
If n total points contain groups of k1, k2, ... collinear points, and no invalid triple is counted twice, then triangles = C(n, 3) - C(k1, 3) - C(k2, 3) - .... For overlapping or more complicated configurations, count each invalid selection carefully.
Diagonals and Segments of a Polygon
Every pair of vertices determines one segment. Adjacent vertex pairs determine the n sides, while the remaining pairs determine diagonals. Therefore, diagonals = C(n, 2) - n = n(n-3)/2.
Chords and Regions in a Circle
The region formula counts the initial region, the chords themselves, and the interior intersections created by choosing four distinct circle points. It is valid only when no three chords intersect at one interior point.
Selecting Sides or Diagonals
If the question only asks for a group of segments, use combinations. If the selected segments must be non-adjacent, intersecting, parallel or form a particular figure, first identify and exclude selections that violate that condition.
Geometry Based Permutation and Combination Visual Guide & Data
Use the table and chart below to understand the data better.
Common Geometry-Based Counting Results
Formulas for selecting points and counting polygon segments, diagonals and circle regions.
| Geometric object | Formula | Condition |
|---|---|---|
| Triangles from n points | C(n, 3) | No three selected points are collinear |
| Segments joining n points | C(n, 2) | Each segment has two distinct endpoints |
| Diagonals of an n-gon | n(n-3)/2 | Polygon has n vertices |
| Chords from n circle points | C(n, 2) | Each chord joins two circle points |
| Maximum circle regions | 1 + C(n, 2) + C(n, 4) | No three chords are concurrent inside |
Note: For collinear points, subtract selections that do not form the required geometric figure.
Hexagon vertices and diagonals
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Geometry Based Permutation and Combination Revision Points
Use combinations for geometric selections because the order of selected points or segments generally does not matter.
- Three non-collinear points form one triangle: C(n, 3).
- If k points are collinear, subtract C(k, 3) invalid triangle selections.
- Total segments joining n points: C(n, 2).
- Diagonals of an n-gon: C(n, 2) - n = n(n-3)/2.
- Chords joining n points on a circle: C(n, 2).
- Maximum regions from all chords: 1 + C(n, 2) + C(n, 4), provided no three chords are concurrent inside.
- Check whether the question asks for all selections or only selections satisfying conditions such as non-collinearity, non-adjacency or intersection.
Geometry Based Permutation and Combination FAQs
How many triangles can be formed from 10 points when no three points are collinear?
The number is C(10, 3) = 10 × 9 × 8 ÷ 6 = 120 triangles.
How many triangles are formed by the vertices of a 12-sided polygon?
Since no three vertices of a polygon are collinear, the number is C(12, 3) = 220.
What is the geometry based permutation and combination formula for diagonals?
For an n-sided polygon, diagonals = C(n, 2) - n = n(n-3)/2. The subtraction removes the n sides from all pairs of vertices.
How many diagonals does a 15-sided polygon have?
Diagonals = 15(15-3)/2 = 15 × 12 ÷ 2 = 90.
How many triangles can be formed from 8 points if 3 points are collinear and no other three are collinear?
Start with C(8, 3) = 56 and subtract the invalid collinear selection C(3, 3) = 1. Therefore, 55 triangles are formed.
How many chords can be drawn using 7 points on a circle?
Each chord is determined by two points, so the number is C(7, 2) = 21.