समीकरण x + y + z = 6 को कितने त्रिक (x, y, z) संतुष्ट करते हैं, जहाँ x, y और z धनपूर्ण संख्याएँ हैं ?
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How many triplets (x, y, z) satisfy the equation x + y + z = 6, where x, y and z are natural numbers?
(a) 4 · (b) 5 · (c) 9 · (d) 10
उत्तर और व्याख्या · explanation in English
Answer: (d) 10
Each of x, y and z is at least 1, so we split 6 units into three positive parts: 5C2 = 10 ordered triplets.
- Natural numbers start from 1, so x, y, z ≥ 1.
- Line up 6 units and choose 2 of the 5 gaps between them to cut into three positive parts: 5C2 = 10.
- Check by listing: (4, 1, 1) in 3 orders, (3, 2, 1) in 6 orders, (2, 2, 2) in 1 order → 3 + 6 + 1 = 10.
Remember · Positive solutions of x₁ + … + xₖ = n number (n − 1)C(k − 1); order matters in triplets.
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). ·