Pigeonhole principle drawing balls from a box
Question
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:
Which of the above statements is/are correct?
Options
1 only
2 only
Both 1 and 2
Neither 1 nor 2
Explanation
Apply the Pigeonhole Principle to evaluate both worst-case scenario thresholds independently:
Statement 1 asks for a guaranteed full group of at least one color. To maximize the delay, assume we draw exactly 1 ball less than the full amount for every single color category available: Max incomplete draws = (14-1) + (20-1) + (26-1) + (28-1) + (38-1) + (54-1) = 13 + 19 + 25 + 27 + 37 + 53 = 174.
The very next ball drawn (174 + 1 = 175) must complete at least one of these color groups. Statement 1 is correct.
Statement 2 asks to guarantee at least one ball of each color. To maximize failure, assume we completely exhaust all the largest color groups first, leaving the single smallest group untouched: Max draws missing a color = 54 (white) + 38 (red) + 28 (yellow) + 26 (green) + 20 (blue) = 166.
The very next selection (166 + 1 = 167) is guaranteed to pull a ball from the remaining black group, ensuring all colors are represented. Statement 2 is correct.
Answer: (c).
Question details
Year
2023
Paper
CSAT
Question
Q36
Section
Logical & Analytical Reasoning
Sub-topic
Probability & Pigeonhole
Type
Inequality logic
Difficulty
Hard
Source hint
Logical deduction
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