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Q41. The smallest square floor which can be completely paved with tiles of size 8x6 without breaking any tile, needs n tiles. Find n.
To form a square, the side length must be a common multiple of the tile's dimensions (8 and 6). The Least Common Multiple (LCM) of 8 and 6 is 24. So, the smallest square floor is 24x24. The area of the floor is 576. The area of one tile is 48. The number of tiles (n) = 576 / 48 = 12.
Q42. A plate of 5m x 2m size with uniform thickness, weighing 20 kg, is perforated with 1000 holes of 5cm x 2cm size. What is the weight of the plate (in kg) after perforation?
The initial area of the plate is 500 cm x 200 cm = 100,000 cm^2. The area of one hole is 5 cm x 2 cm = 10 cm^2. 1000 holes remove an area of 10,000 cm^2. This is exactly 10% of the total area. Removing 10% of the area removes 10% of the weight (2 kg). The remaining weight is 20 - 2 = 18 kg.
Q43. Three boxes are coloured red, blue and green and so are three balls. In how many ways can one put the balls one in each box such that no ball goes into the box of its own colour?
This is a derangement problem for 3 items (d3). The only possible configurations are (R-ball in B-box, B-ball in G-box, G-ball in R-box) and (R-ball in G-box, B-ball in R-box, G-ball in B-box). There are exactly 2 ways.