UPSC Prelims 2026 CSAT, question 23

Topic: Counting and probability

Q23

There are four types of weights, namely 1 kg, 2 kg, 5 kg and 10 kg. What is the maximum number of different ways one can measure 20 kg, if at least eight but not more than eleven weights of 1 kg are to be used while measuring?

(a) 7
(b) 8Correct answer
(c) 9
(d) 10

Solution.

Answer: (b) 8

Let $n_1 \in \{8, 9, 10, 11\}$ be the number of $1$ kg weights used; the remaining weight $R = 20 - n_1 \in \{12, 11, 10, 9\}$ must be formed using $2$ kg, $5$ kg, and $10$ kg weights ( $2n_2 + 5n_5 + 10n_{10} = R$ ).
For $R = 12$ ( $n_1 = 8$ ), there are $3$ ways: $(n_2, n_5, n_{10}) = (6,0,0), (1,2,0), (1,0,1)$ ; for $R = 11$ ( $n_1 = 9$ ), $1$ way: $(3,1,0)$ .
For $R = 10$ ( $n_1 = 10$ ), there are $3$ ways: $(5,0,0), (0,2,0), (0,0,1)$ ; for $R = 9$ ( $n_1 = 11$ ), $1$ way: $(2,1,0)$ , giving $3 + 1 + 3 + 1 = 8$ ways in total.