किसी घन के प्रत्येक फलक को काले या सफेद रंग से रँगा जा सकता है । उस घन को कितने विभिन्न तरीकों से रँगा जा सकता है ?
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Each face of a cube can be painted in black or white colours. In how many different ways can the cube be painted?
(a) 9 · (b) 10 · (c) 11 · (d) 12
उत्तर और व्याख्या · explanation in English
Answer: (b) 10
Treating paintings that can be turned into one another as the same, count by the number of black faces: 0, 1, 5 or 6 black faces give one pattern each, while 2, 3 or 4 give two each. Total 1 + 1 + 2 + 2 + 2 + 1 + 1 = 10.
- Count by the number of black faces; rotations of the cube give the same painting.
- 0 black: 1 way. 1 black: 1 way (any face can be turned to the top).
- 2 black: adjacent or opposite → 2 ways.
- 3 black: all three meeting at a corner, or in a band (two opposite faces plus one between) → 2 ways.
- 4 black = 2 white → 2 ways; 5 black → 1 way; 6 black → 1 way.
- Total = 1 + 1 + 2 + 2 + 2 + 1 + 1 = 10.
Remember · For two-colour cube patterns, count by faces of one colour; k and 6 − k faces give the same count.
Question and answer: UPSC's official GS Paper II (2019, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·