There are 4 horizontal and 4 vertical lines, parallel and equidistant to one another on a board. What is the maximum number of rectangles and squares that can be formed?
- (a) 16
- (b) 24
- (c) 36 ✓ UPSC's answer
- (d) 42
Why the answer is (c)
• A grid formed by 4 horizontal and 4 vertical lines creates a 3x3 array of unit cells, resulting in 9 small squares.
• To form a rectangle or square, one must choose any 2 distinct horizontal lines and any 2 distinct vertical lines from the available sets.
• The number of ways to choose 2 horizontal lines from 4 is calculated as 4C2, which equals 6.
• The number of ways to choose 2 vertical lines from 4 is also 4C2, which equals 6.
• The total number of rectangles and squares is the product of these combinations: 6 * 6 = 36.
• Therefore, the maximum number of rectangles and squares that can be formed is 36.
Why the other options are wrong
- (a) 16
- 16 is the number of unit squares in a 4x4 grid, not the total count of all possible rectangles and squares in a 3x3 grid.
- (b) 24
- 24 is an incorrect calculation that likely results from miscounting the combinations of lines or only counting specific sizes of rectangles.
- (d) 42
- 42 is an incorrect total that does not match the combinatorial calculation of 4C2 * 4C2.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2017, held on 18 June 2017. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.