Raj has ten pairs of red, nine pairs of white and eight pairs of black shoes in a box. If he randomly picks shoes one by one (without replacement) from the box to get a red pair of shoes to wear, what is the maximum number of attempts he has to make ?
- (a) 27
- (b) 36
- (c) 44
- (d) 45 ✓ UPSC's answer
Why the answer is (d)
• Total shoes: 10 red pairs = 20 red shoes, 9 white pairs = 18 white shoes, 8 black pairs = 16 black shoes, so 54 shoes.
• To delay a red pair as long as possible, Raj can first pick all 34 non-red shoes (18 white + 16 black).
• He can then pick one shoe from each of the 10 red pairs, adding 10 red shoes, for 44 picks, still without any complete red pair.
• The 45th shoe must be the matching shoe of one of those 10 red shoes, giving a red pair, so the maximum number of attempts is 45, option (d).
Why the other options are wrong
- (a) 27
- 27 counts only the total number of pairs (10+9+8), not the individual shoes that can be drawn before a red pair is forced.
- (b) 36
- 36 assumes any two red shoes form a pair, ignoring that the 10 red shoes can be one from each distinct red pair.
- (c) 44
- 44 is the maximum number of shoes that can be picked without getting a red pair, not the number needed to guarantee one.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.