Analyzing Box Weights and Total Combinations

The question asks which of the given total weights cannot be formed by combining boxes that weigh 4 kg, 7 kg, and 10 kg. We are given three specific weights: 4 kg, 7 kg, and 10 kg. The phrase "combination of these boxes" typically implies taking a subset of these distinct boxes. We need to find all possible sums we can get by picking zero, one, or more of these three *specific* boxes.

Finding Possible Total Weights

Let the weights of the three boxes be \(w_1 = 4\) kg, \(w_2 = 7\) kg, and \(w_3 = 10\) kg. We can form combinations by choosing a subset of these boxes and summing their weights. The possible subsets and their corresponding total weights are:

• Choosing no boxes: Total weight \(= 0\) kg (empty combination).
• Choosing one box:
  • Box 1: Total weight \(= 4\) kg.
  • Box 2: Total weight \(= 7\) kg.
  • Box 3: Total weight \(= 10\) kg.
• Choosing two boxes:
  • Box 1 and Box 2: Total weight \(= 4 + 7 = 11\) kg.
  • Box 1 and Box 3: Total weight \(= 4 + 10 = 14\) kg.
  • Box 2 and Box 3: Total weight \(= 7 + 10 = 17\) kg.
• Choosing three boxes:
  • Box 1, Box 2, and Box 3: Total weight \(= 4 + 7 + 10 = 21\) kg.

The complete set of possible total weights from combinations of these specific three boxes is \(\{0, 4, 7, 10, 11, 14, 17, 21\}\) kg.

Checking the Given Options

Now, we compare the given options for total weight against the list of possible total weights \(\{0, 4, 7, 10, 11, 14, 17, 21\}\):

• Option 1: $14$ kg. Is $14$ in the list? Yes, \(14 = 4 + 10\). So, 14 kg can be a total weight.
• Option 2: $21$ kg. Is $21$ in the list? Yes, \(21 = 4 + 7 + 10\). So, 21 kg can be a total weight.
• Option 3: $17$ kg. Is $17$ in the list? Yes, \(17 = 7 + 10\). So, 17 kg can be a total weight.
• Option 4: $18$ kg. Is $18$ in the list? No, $18$ is not in the list \(\{0, 4, 7, 10, 11, 14, 17, 21\}\). So, 18 kg cannot be a total weight from combining these three boxes.

Conclusion

Based on the possible combinations of the three given boxes with weights 4 kg, 7 kg, and 10 kg, the total weight of 18 kg cannot be formed.

Revision Table: Box Weight Combinations

Additional Information: Understanding Weight Combination Problems

This type of problem is related to finding subset sums. When you have a specific set of items (like boxes with given weights), a "combination" usually means selecting one or more items from that set. If the problem allowed using multiple boxes of the same weight type (e.g., using two 4 kg boxes), the problem would become a variation of the Change-making problem or Frobenius Coin Problem, where you determine which total values can be formed using a given set of coin denominations (or weights) any number of times. However, the phrasing "combination of these boxes" usually points to using each distinct box at most once unless otherwise specified.

In cases where you can use multiple instances of each weight, you would look for solutions to the equation \(4a + 7b + 10c = W\), where \(a, b, c\) are non-negative integers representing the number of boxes of each weight used, and \(W\) is the target total weight. For the given weights 4, 7, and 10, any sufficiently large integer weight can be formed (since the greatest common divisor of 4, 7, and 10 is 1), but smaller weights may not be possible. However, for this specific problem, the simpler subset sum interpretation is the most fitting given the options and standard problem phrasing.