Permutation and Combination Questions for CAT 2026 — Figure Based Problems with Solutions
- Jul 22, 2025
- 4 min read
Updated: Apr 12
PERMUTAION AND COMBINATION Question for CAT
Q.1 How many straight lines can be formed from 10 points if no three points are collinear?
a.45
b.90
c.10!
d.None of the above
Q.2 There are 10 points in a plane in which 4 are collinear. How many straight lines can be formed from these points?
a.45
b.39
c.40
d.None of the above
Q.3 How many triangles can be formed with 10 points in a plane of which no three points are collinear?
a.10C2
b.10P3
c.10C3
d.None of the above
Q.4 Out of 15 points in a plane, no 3 are in straight line except 8 points which are collinear. How many triangles can be formed by joining them?
a.399
b.15C3 - 8C3 + 1
c.504
d.None of the above
Q.5 There are 25 points on a plane of which 7 are collinear. How many quadrilaterals can be formed from these points?
a.12650
b.11985
c.12020
d.None of the above
Q.6 The number of rectangles on a chess board is
a.144
b.256
c.1296
d.None of the above
Q.7 The number of squares on a chess board is
a.1296
b.204
c.64
d.None of the above
Q.8 In how many ways can a person travel from point A to Point B via the shortest route if the person can travel Only along the edges
a.8!/(4!4!)
b.8P4
c.5C2 x 5C2
d.None of the above
Q.9 The number of rectangles that can be obtained by joining four of the twelve vertices of a 12-sided regular polygon is
a.66
b.30
c.24
d.15
Q.10 At most, how many intersection points are there by drawing 5 straight lines on a plane such that no two lines are overlapping?
a.5C2
b.5P2
c.52
d.25
Q.11 What is the greatest number of points of intersection of 5 straight lines and four circles?
a.60
b.70
c.72
d.62
Q.12 In how many ways, we can choose a black and a white square on a chess board such that the two are not in the same row or column?
a.768
b.1024
c.384
d.None of the above
Q.13 What is the maximum number of regions that 10 straight lines can divide a plane into?
a.45
b.56
c.46
d.None of the above
Q.14 A plane is divided into 79 regions by drawing several straight lines. What is the difference between maximum and minimum number of lines required for the division?
a.78
b.66
c.12
d.None of the above
Q.15 Let AB, CD, EF, GH, and JK be five diameters of a circle with center at O. In how many ways can three points be chosen out of A, B, C, D, E, F, G, H, J, K, and O so as to form a triangle?
a.165
b.160
c.120
d.115
Answer for Permutation and Combination for CAT
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
A | C | C | A | B | C | B | A | D | A |
11 | 12 | 13 | 14 | 15 | |||||
D | A | B | B | B |
Frequently Asked Questions — Figure-Based Permutation Problems for CAT 2026
What are figure-based permutation problems?
Figure-based problems ask: how many triangles, quadrilaterals, diagonals, or other geometric figures can be formed from n given points? They apply combination formulas to geometric contexts. For example: triangles from n non-collinear points = C(n,3).
How many triangles can be formed from n points where some are collinear?
Total triangles from n points = C(n,3). Subtract triangles from collinear points: if k points are collinear, subtract C(k,3). If there are multiple groups of collinear points, subtract C(k_i, 3) for each group i.
How many diagonals does an n-sided polygon have?
Diagonals = C(n,2) - n = n(n-1)/2 - n = n(n-3)/2. For a hexagon (n=6): 6x3/2 = 9 diagonals. This formula subtracts the n sides from all possible line segments C(n,2).
How many quadrilaterals can be formed from n points on a circle?
Any 4 points on a circle form exactly one quadrilateral (a cyclic quadrilateral). So the answer = C(n,4). This applies when all n points are on a circle (no 3 are collinear on any chord).
How do I count regions created by diagonals inside a polygon?
For a convex n-gon with no three diagonals concurrent inside: regions = C(n,4) + C(n,2) - n + 1. This formula counts the regions using Euler's formula for planar graphs: V - E + F = 2.
What is the difference between collinear and non-collinear point problems?
Non-collinear: any 3 points form a triangle. Collinear: 3 points on a line form a line, not a triangle. Always check collinearity. For lines: C(n,2) total lines minus C(k,2) lines from each collinear group plus 1 line per group.
How many straight lines can be drawn through n points where some are collinear?
Total lines = C(n,2) - sum of C(k_i,2) + (number of collinear groups). For example, from 10 points with 4 collinear: C(10,2) - C(4,2) + 1 = 45 - 6 + 1 = 40 lines.
How should I prepare figure-based P&C for CAT?
Memorise the standard formulas: triangles from n points, diagonals of polygon, quadrilaterals on circle. Practice 8-10 problems involving collinear point corrections. These problems appear in CAT every 1-2 years.
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