किसी डिब्बे में 14 काली गेंदें, 20 नीली गेंदें, 26 हरी गेंदें, 28 पीली गेंदें, 38 लाल गेंदें और 54 सफ़ेद गेंदें हैं । निम्नलिखित कथनों पर विचार कीजिए :
- 1.यदि डिब्बे में से यादृच्छिक रूप से कोई सी n गेंदें इस प्रकार निकाली जाएँ कि उनमें कम-से-कम एक रंग का एक पूरा समूह अवश्य हो, तो n की लघुतम संख्या 175 है ।
- 2.यदि डिब्बे में से यादृच्छिक रूप से कोई सी m गेंदें इस प्रकार निकाली जाएँ कि उनमें हर रंग की कम-से-कम एक गेंद अवश्य हो, तो m की लघुतम संख्या 167 है ।
उपर्युक्त कथनों में से कौन-सा/से सही है/हैं ?
Read this question in English
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 · (d) Neither 1 nor 2
उत्तर और व्याख्या · explanation in English
Answer: (c) 1 और 2 दोनों
For a full group, the unluckiest draw takes every colour one ball short of complete: 13 + 19 + 25 + 27 + 37 + 53 = 174, so 175 balls guarantee a full group. For one ball of each colour, the unluckiest draw takes every ball except the smallest colour: 180 − 14 = 166, so 167 balls guarantee all six colours.
- Total balls = 14 + 20 + 26 + 28 + 38 + 54 = 180.
- Statement 1: the worst case leaves every colour one short: 13 + 19 + 25 + 27 + 37 + 53 = 174 balls with no full group. The next ball completes one, so n = 175. Correct.
- Statement 2: the worst case is all balls except the smallest colour (14 black): 180 − 14 = 166 balls without black. The next ball must be black, so m = 167. Correct.
- ✓ 1. 174 balls can still leave every colour one ball short of full; the 175th ball must complete a group.
- ✓ 2. 166 balls can all be non-black; the 167th must be black, giving all six colours.
Remember · Guarantee questions: build the unluckiest draw that still fails, then add one.
Question and answer: UPSC's official GS Paper II (2023, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·