Mathematics

There are 4 men and 4 women, each forming 4 married couples. Teams must be mixed doubles (1 man + 1 woman) without husband-wife pairs. Each man can pair with 3 women (excluding his wife), so 4 men × 3 women = 12 possible teams. The number of ways to form two teams (each with 1 man and 1 woman) without overlap is the number of distinct pairs of teams that can play a game. The total number of such games is 36.

Given the cost of 1 kg flour is Rs.20.50, cost of 6 kg flour = 6 × 20.50 = Rs.123. Since 6 kg flour costs the same as 2 kg rice, cost of 1 kg rice = 123 ÷ 2 = Rs.61.50. The cost of 10 kg apples equals cost of 24 kg rice, so cost of 1 kg apple = (24 × 61.50) ÷ 10 = Rs.147.60. Total cost = (4 × 147.60) + (3 × 61.50) + (5 × 20.50) = 590.40 + 184.50 + 102.50 = Rs.877.40.

Combining like terms on the left side gives 9x - 27, and on the right side 13 - x. Adding x to both sides and adding 27 to both sides results in 10x = 40, so x = 4.

The word 'OPTICAL' has 7 letters with vowels O, I, A and consonants P, T, C, L. Treating the vowels as a single block, we have 5 units to arrange (the vowel block plus 4 consonants). These 5 units can be arranged in 5! = 120 ways. The vowels inside the block can be arranged among themselves in 3! = 6 ways. Therefore, the total number of arrangements where vowels always come together is 5! × 3! = 120 × 6 = 720. This matches option B.

The total number of people is 5 men + 6 women = 11. The number of ways to select 8 persons from 11 is given by the combination C(11,8) = 165. However, since 165 is option A, the original correct answer should be A, not D. The options given do not match the correct calculation except for A. Therefore, the correct answer is A.

Let Aftab's present age be x and Kashif's present age be y. Ten years ago, Aftab's age was (x - 10) and Kashif's age was (y - 10). Given (x - 10) = (1/3)(y - 10) and y = x + 12. Substituting y in the first equation: x - 10 = (1/3)(x + 12 - 10) => x - 10 = (1/3)(x + 2) => 3(x - 10) = x + 2 => 3x - 30 = x + 2 => 2x = 32 => x = 16. So, Aftab's present age is 16 years.

Let the digits be x and y. Given x + y = 12 and |x - y| = 6. Solving these, the digits are 9 and 3. The two-digit number can be 93 or 39. Since the difference is positive 6, the number 93 (9 - 3 = 6) fits the condition. Hence, the answer is 93.

The number of ways to select one trouser and one shirt is the product of the number of trousers and shirts, i.e., 9 × 12 = 108.

The total number of balls in the bag is 9 yellow + 3 white + 4 red = 16 balls. The number of ways to draw two balls from the bag without regard to order is given by the combination formula $\left(\genfrac{}{}{0ex}{}{16}{2}\right)$, which counts the number of ways to choose 2 items from 16. This corresponds to option C (¹⁶C₂). Option D (¹²C₂) is incorrect because the total number of balls is not 12 but 16. Therefore, the correct answer is C.

To find the number of ways five boys and three girls can sit in a row such that all boys sit together, treat the five boys as a single block. Along with the three girls, this gives 4 entities to arrange, which can be done in 4! = 24 ways. The five boys within their block can be arranged among themselves in 5! = 120 ways. Therefore, the total number of arrangements is 4! × 5! = 24 × 120 = 2880. Hence, the correct answer is option C (2880).