Geometrical figures, chessboard and grid is one of the most important and frequently tested topics under permutation and combination in quantitative aptitude. It is asked in almost every competitive exam including CAT, SSC CGL, SSC CHSL, Bank PO, Bank Clerk, Railway RRB and CSAT. A strong understanding of geometrical figures, chessboard and grid concept, formulas and tricks is essential for scoring well in these exams. This post covers number of straight lines from n points, number of straight lines when k points are collinear, number of triangles, number of polygons, number of diagonals of a polygon, chessboard concept, number of rectangles, number of squares and shortest path concept — all explained with clear formulas and solved examples.
Permutation & Combination → Geometrical Figures, Chessboard & Grid
Formation of Geometrical Figures
⦿ Number of straight lines from given n points:
Given, there are ‘n’ points in a plain. no three of them are collinear, then number of straight lines is same as number of ways of selecting 2 points out of ‘n’ points = ⁿC₂
Example:- There are 10 points in a plane. No three of them are collinear. Then how many straight lines can be drawn from these points.
Solution:- ¹⁰C₂ Answer
⦿ Number of straight lines from given ‘n’ points, out of them exactly ‘k’ points are collinear Concept
if no three points would have been collinear then total number of straight lines = ⁿC₂
if out of ‘k’ points no three points would have been collinear then number of straight lines through these ‘k’ points = ᵏC₂
But since these ‘k’ points are collinear so these ‘k’ point will not form ᵏC₂ lines but only 1 line can be made from these ‘k’ collinear points. But in ⁿC₂ calculation these ᵏC₂ lines are also included.
Example:- There are 10 points in a plane. Out of these 4 are collinear. Then how many straight lines can be drawn from these 10 points?
Solution:-
¹⁰C₂ – ⁴C₂ + 1
= 45 – 6 + 1
= 40 Answer
⦿ Number of Triangle from given ‘n’ points is no three of them are collinear = ⁿC₃
Example:- How many triangle can be drawn from 10 points in a plane if no three of them are collinear?
Solution:-
= ¹⁰C₃
= \(\frac{{10!}}{{7!3!}}\)
= 120 Answer
⦿ Number of polygon of ‘r’ sides from given ‘n’ points if no three of them are collinear = ⁿCᵣ
Proof:- A polygon of ‘r’ sides has ‘r’ vertices.
∴ Number of total polygon = number of ways of selecting ‘r’ things out of ‘n’ things
= ⁿCᵣ
Example:- How many octagon can be drawn from 10 points in a plane if no three of them are collinear?
Solution:- ¹⁰C₈
= \(\frac{{10 \times 9}}{2}\)
= 45 Answer
⦿ Number of diagonal of a polygon of n sides
Since diagonal is formed by joining two vertices i.e. by joining two points . Since polygon of n sides has n vertices. Hence total number of straight lines that can be drawn out of these ‘n’ points.
Hence number of diagonal = ⁿC₂ – number of sides
= ⁿC₂ – n
= \(\frac{{n(n – 1)}}{2} – n\)
Example:- What is the number of diagonal of a hexagon?
Solution:- ⁶C₂ – 6
= 9 Answer
Example:- If a polygon has 35 diagonal. Then how many sides that polygon has?
Solution:-
\(\frac{{n(n – 3)}}{2}\) = 35
n² – 3n – 70 = 0
n² -10n + 7n – 70 = 0
n(n-10) + 7(n-10) = 0
(n-10)(n+7) = 0
n=10
∴ polygon has 10 sides Answer
Chess Board Concept
A chess board has 8 rows & 8 columns.
Number of squares in chess board ⇒
Number of square of size 1×1 = 8×8 = 64
Number of square of size 2×2 = 7×7 = 49
Number of square of size 3×3 = 6×6 = 36
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Number of square of size 8×8 = 1×1 = 1
Hence total number of squares in chess board = 1² + 2² + 3² + ………… + 8²
= 204
Number of Rectangle in chess board ⇒
To calculate number of rectangles we have to select 2 lines from 9 verticle lines. Similarly, another set of 2 lines from 9 horizontal lines.
Hence total number of rectangle = ⁹C₂ × ⁹C₂
= 1296
Here point to remember is that rectangle includes squares in calculation
Thus in chess board of size n×n ⇒
Example:- Find the number of squares in chess board of size 12×12
Solution:-
1² + 2² + 3² + 4² + ……….. + 12²
Here n =12
∴ number of squares = \(\frac{{n(n + 1)(2n + 1)}}{6}\)
= \(\frac{{12 \times 13 \times 25}}{6}\)
= 13×50
= 650 Answer
Example:- Find the number of rectangle in chess board of size 12×12.
Solution:-
Here n = 12
∴ number of rectangle = \({\left\{ {\frac{{n(n + 1)}}{2}} \right\}^2}\)
= \({\left\{ {\frac{{12 \times 13}}{2}} \right\}^2}\)
= 78²
= 6084 Answer
Grid based problem Concept
Example:- Find the number of rectangles in 10×12 grid.
Solution:-
¹⁰⁺¹C₂ × ¹²⁺¹C₂
= ¹¹C₂ × ¹³C₂
= 55 × 78
= 4290 Answer
Example:- Find the number of squares in 10×12 grid.
Solution:-
10×12 = 120
9×11 = 99
8×10 = 80
7×9 = 63
6×8 = 48
5×7 = 35
4×6 = 24
3×5 = 15
2×4 = 8
1×3 = 3
________
Total = 495
________
Hence total number of squares = 495 Answer
Shortest Path Concept
Consider a Grid m×n. then we are to find the number of shortest path from one corner to diagonally opposite corner.
Proof with Example:-
Consider a 5×8 Grid. what is the minimum number of ways a person can travel from one corner to diagonally opposite corner?
Since we are to go from (0,0) to (5,8). & since (5,8) is 8 step right & 5 step up from (0,0).
So we move in anyway but to reach (5,8) we have to ultimately travel 8 steps right & 5 steps up.
we can travel these R & U steps in any order but the total steps will always be (8+5) = 13
→ we are to find the number of ways in which we can complete these 13 steps ⇒
In these 13 steps path for a particular horizontal 8 steps there is only 1 option left for 5 verticle steps & vice-versa.
So we are to only select 8 or 5 steps from total 13 steps which will be our final answer.
Hence total number of shortest path = ¹³C₈ = ¹³C₅ Answer