The weights of four boxes are 10 kg, 40 kg, 50 kg and 80 kg. Which of the following cannot be obtained as the total weight, in kilograms, of any combination of these boxes if each box is used at most once?

Introduction / Context:
This is another combination of weights puzzle that checks whether you can accurately list possible totals formed from a set of four distinct weights. The skill is the same: identify which one of the suggested totals cannot appear as a sum of any subset of the given weights when each can be used at most once. It is a useful exercise in both reasoning and organised enumeration.

Given Data / Assumptions:

• Box weights: 10 kg, 40 kg, 50 kg, 80 kg.
• Each box can be included at most one time in a combination.
• Candidate totals: 120 kg, 100 kg, 90 kg, 110 kg.
• No fractions of a box are allowed.

Concept / Approach:
We systematically work out all possible sums from one, two, three and four box combinations. Once we have the set of attainable sums, we simply check which of the given options does not appear in that set. This approach eliminates guesswork and ensures that every possibility has been considered logically.

Step-by-Step Solution:
Step 1: Note the weights: 10, 40, 50, 80.Step 2: Sums using one box: 10, 40, 50, 80.Step 3: Sums using two boxes: 10 + 40 = 50, 10 + 50 = 60, 10 + 80 = 90, 40 + 50 = 90, 40 + 80 = 120, 50 + 80 = 130.Step 4: Sums using three boxes: 10 + 40 + 50 = 100, 10 + 40 + 80 = 130, 10 + 50 + 80 = 140, 40 + 50 + 80 = 170.Step 5: Sum using all four boxes: 10 + 40 + 50 + 80 = 180.Step 6: Collect all distinct sums: 10, 40, 50, 60, 80, 90, 100, 120, 130, 140, 170, 180.Step 7: Compare each option 120, 100, 90 and 110 with this list.

Verification / Alternative check:
We can see that 120 appears as 40 + 80. The total 100 appears as 10 + 40 + 50. The total 90 appears as either 10 + 80 or 40 + 50. However, 110 is not present. To double check, try to construct 110: 80 + 40 = 120 (too large), 80 + 50 = 130 (too large), 80 + 10 = 90 (too small), and 50 + 40 = 90. Adding the 10 kg weight to 100 gives 110, but 100 itself was 10 + 40 + 50, which already uses all three smaller boxes. Using all four boxes gives 180, not 110. Therefore, 110 cannot be formed.

Why Other Options Are Wrong:
120 kg is attainable from 40 + 80.100 kg is attainable from 10 + 40 + 50.90 kg is attainable from 10 + 80 or from 40 + 50.These values are therefore not impossible totals and so cannot be the correct answer to the question.

Common Pitfalls:
A common error is to overlook a combination like 10 + 80 or 10 + 40 + 50 and thus wrongly conclude that 90 or 100 cannot be formed. Another mistake is to attempt to solve the problem mentally without writing down intermediate sums, which increases the chance of missing a valid subset. A structured list, as shown, is the best way to avoid such oversights.

Final Answer:
The total weight that cannot be obtained from any combination of the four boxes is 110 kilograms.