UPSC Prelims 2009 Q95: Science & Technology – Quantitative Aptitude
A person has 4 coins each of different denomination. What is the number of different sums
of money the person can form (using one or more coins at a time)?
- (a) 16
- (b) 15
- (c) 12
- (d) 11
Answer: (b) 15
Explanation
Counting distinct sums from four coins of different denomination
A person has 4 coins, each of a different denomination — we need the number of distinct sums possible using one or more at a time.
| Step | Calculation |
|---|---|
| Total non-empty subsets of 4 coins | 2⁴ − 1 = 15 |
| Check for duplicate sums | Since all four coins have genuinely distinct values, each subset combination produces a unique total |
| Conclusion | All 15 non-empty subsets give 15 distinct sums |
Why no duplicates occur:
| Point | Detail |
|---|---|
| Four distinct denominations | No two different subsets happen to add up to the same total |
| Formula | For n distinct-value items with no coinciding sums: 2ⁿ − 1 distinct non-empty sums |
Answer: (b) 15.
Why each option is right or wrong
- (a) Incorrect. 16 would include the empty subset (using zero coins), which isn't a valid 'sum of money' formed.
- (b) CORRECT. All 15 non-empty subsets of 4 distinct-denomination coins produce distinct sums.
- (c) Incorrect. 12 undercounts the actual number of distinct possible sums.
- (d) Incorrect. 11 also undercounts the total possible distinct sums.
Key point
Four coins of different denominations produce 2⁴ − 1 = 15 distinct non-empty subset sums, since no two different combinations happen to add up to the same total.
Remember
4 coins of different denominations: 2⁴−1 = 15 distinct possible sums.
Common mistakes
- Including the empty subset (using zero coins), which doesn't count as 'forming a sum of money'.
- Assuming some subset sums might coincide, without confirming that these specific distinct denominations avoid overlaps.
- Using an incorrect formula instead of the simple 2ⁿ − 1 non-empty-subset count.
Exam tip
With n distinct-value 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)