Suppose you have sufficient amount of rupee currency in three denominations : ₹ 1, ₹ 10 and ₹ 50. In how many different ways can you pay a bill of ₹ 107 ?

(a) 16
(b) 17
(c) 18 ✓ UPSC's answer
(d) 19

Why the answer is (c)

• Let the bill be paid with x notes of ₹ 50 and y notes of ₹ 10, with the rest in ₹ 1; the ₹ 1 part is fixed once x and y are chosen, so we count the pairs with 50x + 10y ≤ 107.

• x = 0: y can be 0 to 10, giving 11 ways.

• x = 1: 10y ≤ 57, so y can be 0 to 5, giving 6 ways.

• x = 2: 10y ≤ 7, so y = 0, giving 1 way.

• Total = 11 + 6 + 1 = 18 ways, option (c).

Why the other options are wrong

(a) 16: 16 misses two of the 18 combinations counted above.
(b) 17: 17 misses one of the 18 combinations counted above.
(d) 19: 19 counts one combination more than actually exists.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2019, held on 2 June 2019. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.