The weights of four boxes are 30, 70, 60 and 20 kilograms. Which of the following cannot be the total weight, in kilograms, of any combination of these boxes if in a combination each box can be used at most once?

Introduction / Context:
This problem is about subset sums. We are given the weights of four different boxes and asked which given total weight cannot be made by selecting some or all of the boxes, with the restriction that each box can be used at most once in a combination. This is a common type of question in aptitude tests that checks systematic reasoning and combinatorial thinking.

Given Data / Assumptions:

• The four box weights are 30 kilograms, 70 kilograms, 60 kilograms and 20 kilograms.
• Each combination can use any subset of these boxes, including possibly only one box or all four boxes.
• No box may be used more than once in any combination.
• We are given four candidate totals: 180, 190, 120 and 150 kilograms.
• We must find which total cannot be formed by adding the weights of some subset of the four boxes.

Concept / Approach:
The straightforward method is to list possible sums of subsets of the four box weights. Since there are only four boxes, the number of subsets is small. We can compute all distinct sums that can arise and then compare them with the candidate totals. The total that does not appear in our list of achievable sums is the required answer.

Step-by-Step Solution:
Step 1: Let the weights be 30, 70, 60 and 20 kilograms.Step 2: First consider combinations of one box: the possible totals are 30, 70, 60 and 20.Step 3: Consider combinations of two boxes: 30 + 70 = 100, 30 + 60 = 90, 30 + 20 = 50, 70 + 60 = 130, 70 + 20 = 90, and 60 + 20 = 80. The distinct two box sums are 50, 80, 90, 100 and 130.Step 4: Consider combinations of three boxes: 30 + 70 + 60 = 160, 30 + 70 + 20 = 120, and 30 + 60 + 20 = 110, while 70 + 60 + 20 = 150. The three box sums are 160, 120, 110 and 150.Step 5: Consider the combination of all four boxes: 30 + 70 + 60 + 20 = 180.Step 6: Collect all distinct possible totals: 20, 30, 50, 60, 70, 80, 90, 100, 110, 120, 130, 150, 160 and 180.Step 7: Compare these with the options: 180 appears, 120 appears, 150 appears, but 190 does not appear in the list.Step 8: Therefore 190 kilograms cannot be formed with any combination of the given boxes.

Verification / Alternative check:
We can also reason more directly about the maximum possible weight and near maximum weights. The sum of all four boxes is 180 kilograms, so no combination can exceed 180. Therefore any option greater than 180, such as 190, is immediately impossible. This gives a very fast check. The other totals 120 and 150 are less than 180, so they could potentially be sums. From the earlier subset calculations, we see that 120 and 150 are indeed achievable, while 190 is not. This cross checks our conclusion.

Why Other Options Are Wrong:
The total 180 is obtained by using all four boxes: 30 + 70 + 60 + 20 = 180. The total 120 is obtained by the combination 30 + 70 + 20. The total 150 is obtained by the combination 70 + 60 + 20. Because each of these totals matches at least one valid subset of boxes, they can be formed and therefore are not correct answers to the question.

Common Pitfalls:
One common mistake is to overlook the possibility of using all four boxes and conclude that a large total like 180 or 150 cannot be made. Another mistake is careless arithmetic that causes either double counting or missing some subset. A calm, systematic listing of subsets, or a quick look at the maximum possible sum, helps avoid such errors.

Final Answer:
The only total that cannot be obtained from any combination of the four boxes is 190 kilograms.