Counting in Geometry

(a) In a plane, if there are n points of which no three are collinear, then

1. The number of straight lines that can be formed by joining them is ⁿC₂.
2. The number of triangles that can be formed by joining them is ⁿC₃.
3. The number of polygons with k sides that can be formed by joining them is ⁿC_k.

(b) In a plane, if there are n points out of which m points are collinear, then

1. The number of straight lines that can be formed by joining them is ⁿC₂ – ^mC₂ + 1.
2. The number of triangles that can be formed by joining them is ⁿC₃ – ^mC₃.
3. The number of polygons with k sides that can be formed by joining them is ⁿC_k – ^mC_k.

(c) The number of diagonals of an n sided polygon is ⁿC₂ – n = n × (n – 3)/2

(d) The number of triangles that can be formed by joining the vertices of an n-sided polygon which has,

1. Exactly one side common with that of the polygon is n × (n – 4).
2. Exactly two sides common with that of the polygon are n.
3. No side common with that of the polygon is n × (n – 4) × (n – 5)/6.

(e) The number of parallelograms formed if ‘x’ lines in a plane are intersected by ‘y’ parallel lines is x × y × (x – 1) × (y – 1)/4.

(f) If there are n lines drawn in a plane such that no two of them are parallel and no three of them are concurrent, then the number of different points at which these lines will intersect each other is ⁿC₂ = n × (n – 1)/2.

(g) If there are n straight lines in a plane with no two of them parallel to each other, no three passing through the same point and  their points of intersection are joined, then the number of new lines obtained are n × (n – 1) × (n – 2) × (n – 3)/8.

(h) The sides of a triangle a, b and c are integers where a ≤ b ≤ c. If c is given, then the number of different triangles is c × (c + 2)/4 or (c + 1)2/4, depending on whether c is even or odd. Also, the number of isosceles triangles is (3c – 2)/2 or (3c – 1)/2, depending on whether c is even or odd.

(i) In a square grid  of n × n,

1. The number of rectangles of any size is ∑r³.
2. The number of squares of any size is ∑r².

(j) In a rectangular grid  of p × q (p < q),

1. The number of rectangles of any size is p × q × (p + 1) × (q + 1) / 4.
2. The number of squares of any size is ∑ [(p + 1 – r) × (q + 1 – r)].

(k) If n straight lines are drawn in a plane such that no two lines are parallel and no three lines are concurrent, then the number of parts these lines divide the plane into is equal to [n × (n + 1)/2] + 1.

(l) If “n” parallel lines are passing through a circle, dividing the plane into distinct non-overlapping bounded or unbounded regions, then the maximum number of regions the plane can be divided into is (3n + 1).

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Quant Question Of The Day : 18/06/2018
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