1) 1/32 On the sixth floor there are 2^5 = 32 vents since there is 1 vent on the first floor and it doubles each subsequent floor. This makes the probability of him randomly choosing the correct path 1/32 2) 900 The volume of any cube or rectangular prism is the product of all of its sides. If l stands for the length of the guard's room, w stands for the width of the guard's room, and h stands for the height of the guard's room: l x w x h = 27,000. The volume of the new room can be found by the equation: (h x 1/5) x (l x 1/3) x (w x 1/2) = (l x w x h) x 1/5 x 1/3 x 1/2 (l x w x h) = 27,000 so the answer is 27,000 x 1/5 x 1/3 x 1/2 = 900 3) 5184 The largest four-digit palindrome is 9999 since it is the same forwards and backwards. The next smallest palindrome is 9 x 8 x 8 x 9 = 5184 4) 40 The most intuitive way is to imagine that there are 5 lasers on the floor and 5 on the ceiling. This would mean there is 1 ceiling laser turned on and 3 floor lasers turned on, adding to 4 lasers. Since there are 10 lasers in total, the percent of lasers that are turned on is (4/10) x 100 = 40% The simplest way mathematically is to take the average of the two fractions: (3/5 + 1/5) / 2 = 2/5 = 40%