A box contains 14 black balls, 20 blue balls, 26 green balls, 28 yellow balls, 38 red balls and 54 white balls. Consider the following statements : 1. The smallest number n such that any n balls drawn from the box randomly must contain one full group of at least one colour is 175. 2. The smallest number m such that any m balls drawn from the box randomly must contain at least one ball of each colour is 167. Which of the above statements is/are correct ?
- (a) 1 only
- (b) 2 only
- (c) Both 1 and 2 ✓ UPSC's answer
- (d) Neither 1 nor 2
Why the answer is (c)
• To find the smallest n that guarantees a full group of at least one colour, we first find the maximum number of balls that can be drawn while still missing at least one ball of every colour: 13 black, 19 blue, 25 green, 27 yellow, 37 red, and 53 white, giving 13 + 19 + 25 + 27 + 37 + 53 = 174.
• Adding one more ball to 174 forces at least one colour to be completely represented, so n = 174 + 1 = 175, confirming Statement 1.
• To find the smallest m that guarantees at least one ball of each of the six colours, we find the maximum number of balls that can be drawn while omitting at least one colour entirely.
• To maximise the draw while missing a colour, we omit the colour with the fewest balls, which is black (14), allowing us to draw all remaining 180 − 14 = 166 balls without including any black ball.
• Adding one more ball to 166 forces the presence of a black ball, completing the set of all six colours, so m = 166 + 1 = 167, confirming Statement 2.
• Since both statements are correct, the answer is option (c).
Why the other options are wrong
- (a) 1 only
- Statement 2 is also correct: the worst case for missing a colour is omitting the 14 black balls, giving 166, so m = 167, making Statement 2 valid as well.
- (b) 2 only
- Statement 1 is also correct: the worst case for avoiding a full group is drawing one fewer of each colour (13+19+25+27+37+53 = 174), so n = 175, making Statement 1 valid as well.
- (d) Neither 1 nor 2
- Both statements are correct; Statement 1 gives n = 175 and Statement 2 gives m = 167, so it is not the case that neither is correct.
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.