Question 1

How many distinct paths of length 3 (moving only right or down) exist from the top-left corner to the bottom-right corner of a 3×3 grid?

A) 6
B) 10
C) 15
D) 20

Accepted Answer

In a 3×3 grid, to go from top-left to bottom-right requires exactly 2 moves right (R) and 2 moves down (D), totaling 4 moves. However, the question specifies 'length 3', which is ambiguous. Interpreting as 3 unit segments: we need 2R and 1D (or 1R and 2D). For 2R and 1D: arrangements = C(3,1) = 3. For 1R and 2D: arrangements = C(3,2) = 3. Total = 6. But if 'length 3' means 3 steps to reach a position 3 units away, we need 2R+1D or 1R+2D, giving C(3,2)+C(3,1) = 3+3 = 6. However, standard interpretation for a 3×3 grid (4 moves needed) with 'length 3' paths might mean paths of exactly 3 edges, reaching intermediate positions. Clarifying: paths of 3 edges from (0,0) to (3,0) or (0,3) or (2,1) or (1,2). Most likely: C(3,2) = 3 for RRD, C(3,1) = 3 for RDD, total = 6. But if the grid is 4×4 and we need length 3: C(3,1)=3 for RRD, C(3,2)=3 for RDD, total = 6. Reconsidering: for a standard 3×3 grid (3 units × 3 units = 4×4 vertices), reaching bottom-right requires 3R and 3D (6 moves). Paths of length 3 would reach positions like (3,0), (2,1), (1,2), (0,3). Total such paths = C(3,1)+C(3,2)+C(3,3)+C(3,0) = 3+3+1+1 = 8. Most reasonable: C(6,3) = 20 total paths to bottom-right; if 'length 3' is a constraint, answer is likely 20 (all shortest paths).

Question 2

A figure consists of a rectangle with 5 vertical lines and 4 horizontal lines drawn inside it (including the borders). These lines create a grid of smaller rectangles. Additionally, 6 of these smaller rectangles are shaded. How many total rectangles of all possible sizes can be counted in this figure, excluding the shaded ones?

Accepted Answer

This tests the ability to count all possible rectangles in a grid (not just the unit rectangles) while applying an exclusion constraint.

Question 3

A figure shows a hexagon with all 6 sides equal. Inside, there are 2 diagonals drawn from one vertex to two non-adjacent vertices. Additionally, the hexagon is divided into 3 equal triangular regions by drawing lines from the center to each alternate vertex. How many total triangles of all sizes can be counted in this figure?

Accepted Answer

This tests counting triangles in a complex figure with multiple overlapping divisions and internal structures.

Question 4

In a complex figure, there is a large square divided into 4 equal smaller squares. Each of these 4 squares is further divided into 4 equal smaller squares. Additionally, there are 3 circles inscribed in 3 of the smallest squares, and 2 triangles inscribed in 2 other smallest squares. How many total geometric shapes (squares of all sizes + circles + triangles) are present in the figure?

Accepted Answer

This question tests the ability to count overlapping and nested figures systematically without missing any layer or double-counting.

Question 5

A square is divided into 9 equal smaller squares (3×3 grid). In 4 of these smaller squares, circles are inscribed. In 2 other smaller squares, squares are inscribed. The remaining 3 smaller squares are left empty. How many total distinct shapes (all sizes of squares + circles + inscribed squares + empty regions) are present in the figure?

Accepted Answer

This tests the ability to count all distinct geometric entities in a composite figure, including nested shapes and empty regions.

Question 6

A figure consists of a triangle with all three sides equal (equilateral). From each vertex, a line is drawn to the midpoint of the opposite side. These three lines (medians) intersect at a single point (centroid), dividing the triangle into 6 smaller triangles. Additionally, 2 of these 6 smaller triangles are shaded. How many total triangles of all sizes can be counted in this figure, including both shaded and unshaded triangles?

Accepted Answer

This tests counting triangles in a figure with medians creating multiple overlapping triangular regions, while managing the shading constraint.

Question 7

A rectangle is divided by 3 vertical lines and 2 horizontal lines (not including the borders) into smaller rectangular cells. Additionally, 4 of these cells are further subdivided by a diagonal line each. How many total triangles can be counted in this figure?

Accepted Answer

This tests the ability to count triangles created by diagonal lines within a grid of rectangular cells, combining grid-based counting with triangle enumeration.

SSC MTS Counting Figures

Counting Figures— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Counting Figures — Practice Questions

60-Second Revision — Counting Figures