UPSC Prelims 2010 Q40: Science & Technology – Quantitative Aptitude
A person X has four notes of Rupee 1, 2, 5 and 10 denomination. The number of different
sums of money she can form from them is
- (a) 16
- (b) 15
- (c) 12
- (d) 8
Answer: (b) 15
Explanation
Counting distinct sums from four distinct-value notes
Person X has one note each of ₹1, ₹2, ₹5, and ₹10 — we need the number of distinct sums possible from any combination of these notes.
| Step | Calculation |
|---|---|
| Total non-empty subsets of 4 notes | 2⁴ − 1 = 15 |
| Check for duplicate sums | With denominations 1, 2, 5, 10 (quite spread apart), every subset produces a different sum — no two different subsets add up to the same total |
| Conclusion | All 15 non-empty subsets give 15 distinct sums |
Why no duplicates occur here:
| Point | Detail |
|---|---|
| Denominations | 1, 2, 5, 10 — sufficiently spread out |
| Subset sums | Each combination's total is unique |
Answer: (b) 15.
Why each option is right or wrong
- (a) Incorrect. 16 would include the empty subset (sum = 0), which isn't a valid 'sum of money' formed.
- (b) CORRECT. All 15 non-empty subsets of {1, 2, 5, 10} produce distinct sums.
- (c) Incorrect. 12 undercounts the actual number of distinct non-empty subset sums.
- (d) Incorrect. 8 significantly undercounts the total possible distinct sums.
Key point
Four distinct-value notes (₹1, ₹2, ₹5, ₹10) produce 2⁴ − 1 = 15 distinct non-empty subset sums, since no two different combinations happen to add up to the same total.
Remember
Four notes of ₹1, 2, 5, 10: 2⁴−1 = 15 distinct possible sums of money.
Common mistakes
- Including the empty subset (sum = 0), which doesn't count as 'forming a sum of money'.
- Assuming some subset sums might coincide, without checking that these specific denominations don't overlap.
- Using an incorrect formula instead of the simple 2ⁿ − 1 non-empty-subset count.
Exam tip
With n distinct items where no subset sums coincide, the number of distinct non-empty sums is 2ⁿ − 1. Here, n=4 gives 15.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)