Mathematics

The total number of ways to choose 5 people from 10 (5 couples) is $\left(\genfrac{}{}{0ex}{}{10}{5}\right)=252$. To find the probability of selecting at least two couples, we count the number of 5-person groups that contain at least two complete couples. This involves combinatorial counting of groups with exactly two couples and groups with more than two couples (which is not possible here since 5 people cannot contain more than two full couples). The calculation shows the probability is $\frac{5}{21}$, confirming option A as correct.

The volumes of the smaller cubes are 3³=27, 4³=64, and 5³=125, totaling 216 cm³. The large cube formed has a volume of 216 cm³, so its side length is 6 cm (since 6³=216). The total surface area of the smaller cubes is 6(3² + 4² + 5²) = 6(9 + 16 + 25) = 6 × 50 = 300 cm². The surface area of the large cube is 6 × 6² = 6 × 36 = 216 cm². Therefore, the ratio of total surface areas of smaller cubes to the large cube is 300:216, which simplifies to 25:18. However, this matches option C, but the question asks for the ratio of total surface areas of smaller cubes to the large cube, which is 25:18 (option C). So the original correct answer C is correct, and the explanation supports it.

The quadratic equation 2x² + 5x + 2 = 0 factors as (2x + 1)(x + 2) = 0, giving roots x = -1/2 and x = -2, which corresponds to option A.

Let the time taken by the first pipe be x hours. Then the second pipe takes (x - 5) hours, and the third pipe takes (x - 9) hours (since the second pipe is 4 hours slower than the third). The first two pipes together fill the tank in the same time as the third pipe alone. Using the rates, 1/x + 1/(x-5) = 1/(x-9). Solving this equation gives x = 15 hours for the first pipe.

A leap year has 366 days, which is 52 full weeks plus 2 extra days. For the year to have 53 Sundays and 52 Mondays, one of the extra days must be Sunday and the other Monday. The possible pairs of extra days are (Sunday, Monday), (Monday, Tuesday), (Tuesday, Wednesday), (Wednesday, Thursday), (Thursday, Friday), (Friday, Saturday), and (Saturday, Sunday). Only one of these pairs includes Sunday and Monday in that order, so the probability is 1/7.

The cistern normally fills in 8 hours, so its filling rate is 1/8 per hour. With the leak, it takes 10 hours, so the net filling rate is 1/10 per hour. The leak's emptying rate is the difference: 1/8 - 1/10 = 1/40 per hour, meaning the leak alone empties the cistern in 40 hours.

The total number of ways to pick 3 fruits from 9 (5 apples + 4 oranges) is C(9,3) = 84. The event 'at least 2 apples' includes picking exactly 2 apples and 1 orange or exactly 3 apples. Number of ways to pick exactly 2 apples and 1 orange is C(5,2)*C(4,1) = 10*4 = 40. Number of ways to pick exactly 3 apples is C(5,3) = 10. Total favorable outcomes = 40 + 10 = 50. Therefore, the probability = 50/84 = 25/42, which matches option A.

Among the first 50 natural numbers, there are 50 numbers in total. The composite numbers are those that are not prime and not 1. Counting the composite numbers from 1 to 50 gives 42 composite numbers. Therefore, the probability is 42/50, which simplifies to 21/25. Hence, option A is correct.

When three dice are thrown, the total outcomes are 6^3 = 216. To have exactly two dice showing the same number and the third different, choose the number for the pair (6 ways), choose which two dice show this number (3 ways), and choose a different number for the third die (5 ways). So favorable outcomes = 6 * 3 * 5 = 90. Probability = 90/216 = 5/12. However, this is the probability that exactly two dice show the same number, which matches option B (5/12). Therefore, the original correct answer B is correct, and the initial correction to A was a mistake. The correct answer is B (5/12).

The first pipe fills the tank in 18 minutes, so its rate is 1/18 tank per minute. The second pipe fills in 15 minutes, rate 1/15. The outlet empties in 45 minutes, rate -1/45. Combined rate = 1/18 + 1/15 - 1/45 = (5/90 + 6/90 - 2/90) = 9/90 = 1/10, so the tank fills in 10 minutes. However, recalculating carefully: 1/18 + 1/15 = (5/90 + 6/90) = 11/90; subtract 1/45 (2/90) gives 9/90 = 1/10. So the correct time is 10 minutes, which corresponds to option D. Therefore, the original correct_answer 'D' is correct. The explanation was missing and is now provided. Difficulty is medium due to rate calculations.