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?

(a) 16 ✓ UPSC's answer
(b) 15
(c) 14
(d) 10

Why the answer is (a)

• There are 5 denominations: ₹50, ₹100, ₹200, ₹500 and ₹2,000.

• Taking at least three denominations at a time means choosing subsets of size 3, 4 or 5.

• The number of such subsets is 5C3 + 5C4 + 5C5 = 10 + 5 + 1 = 16.

• Each denomination is greater than the sum of all smaller denominations: 100 > 50, 200 > 150, 500 > 350, 2000 > 850.

• Therefore no two different subsets give the same sum, so all 16 subset sums are different, making the answer (a) 16.

Why the other options are wrong

(b) 15: Option (b) 15 is one less than the required 16 because it omits one valid subset sum from the 10 three-denomination, 5 four-denomination and 1 five-denomination combinations.
(c) 14: Option (c) 14 is two less than the required 16 because it omits two valid subset sums from the 10 three-denomination, 5 four-denomination and 1 five-denomination combinations.
(d) 10: Option (d) 10 counts only the exactly-three-denomination cases, 5C3, and ignores the 5 four-denomination and 1 five-denomination sums.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2020, held on 4 October 2020. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.