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?
Detailed Explanation:
We need a + 2b + 5c + 10d = 20 where a ∈ {8, 9, 10, 11} (1 kg weights) and b, c, d ≥ 0 (2 kg, 5 kg, 10 kg weights).
Case a = 8: 2b + 5c + 10d = 12 → (b,c,d) = (6,0,0), (1,2,0), (1,0,1) → 3 ways
Case a = 9: 2b + 5c + 10d = 11 (odd, so c odd) → (b,c,d) = (3,1,0) → 1 way
Case a = 10: 2b + 5c + 10d = 10 → (b,c,d) = (5,0,0), (0,2,0), (0,0,1) → 3 ways
Case a = 11: 2b + 5c + 10d = 9 (odd, so c odd) → (b,c,d) = (2,1,0) → 1 way
Therefore, total = 3 + 1 + 3 + 1 = 8, making Option 2 the correct answer.
Question 7 of 7 HCF and LCM
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