How many different sums can be formed with the denominations ₹ 50, ₹ 100, ₹ 200, ₹ 500 and ₹ 2,000 taking at least three denominations at a time?
Answer & explanation
Answer: (a) 16
Choosing 3, 4 or all 5 of the five denominations gives 10 + 5 + 1 = 16 selections. Each denomination is more than the sum of all the smaller ones, so no two selections can give the same amount — all 16 sums are different.
- Ways to choose 3, 4 or 5 of 5 denominations: ⁵C₃ + ⁵C₄ + ⁵C₅ = 10 + 5 + 1 = 16.
- 100 > 50, 200 > 50 + 100, 500 > 50 + 100 + 200 and 2,000 > 850, so different selections always give different totals.
- Number of different sums = 16.
- Check: the 16 sums run from ₹ 350 (50 + 100 + 200) to ₹ 2,850 (all five) with no repeats.
Remember · Count selections with nCr, then check whether two selections can give the same total before answering.
Question and answer: UPSC's official GS Paper II (2020, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·