Correct Option (D)
The problem requires determining the maximum number of attempts to obtain a red pair of shoes, which corresponds to a worst-case scenario.
- Total shoes available:
- Red shoes: 10 pairs × 2 = 20 shoes
- White shoes: 9 pairs × 2 = 18 shoes
- Black shoes: 8 pairs × 2 = 16 shoes
- In the worst-case, Raj would first pick all shoes that are not red.
- He picks all 18 white shoes.
- He picks all 16 black shoes.
- Total non-red shoes picked = 18 + 16 = 34 shoes.
- At this stage, no red pair has been formed, and only red shoes remain in the box (20 shoes).
- To further maximize the number of attempts before forming a red pair, Raj would then pick one shoe from each of the 10 red pairs. This results in 10 individual red shoes, none of which form a pair with each other (e.g., 10 left shoes).
- Total attempts made so far = 34 (non-red shoes) + 10 (single red shoes) = 44 attempts.
- After 44 attempts, Raj possesses 10 single red shoes, and the remaining 10 red shoes in the box are their respective matches.
- The next attempt (the 45th pick) will inevitably complete a red pair, as any red shoe picked will match one of the 10 single red shoes already held.
- Therefore, the maximum number of attempts required is 45.
Incorrect Options:
27: This value typically arises from summing the number of pairs (10 + 9 + 8 = 27), which does not account for individual shoes or the worst-case scenario for forming a pair.
36: This value might result from considering all non-red shoes (34) plus a small number of red shoes, but it fails to incorporate the worst-case strategy of picking single red shoes to delay forming a pair.
44: This represents the number of attempts made before a red pair is guaranteed. It is the point where all non-red shoes and one shoe from each red pair have been picked. The 45th attempt is required to complete the pair, making 44 an incomplete answer for the maximum attempts to get a pair.