In Arun's family, three people give their opinions about his weight. Arun thinks his weight is greater than 65 kg but less than 72 kg. His brother thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother believes that his weight cannot be greater than 68 kg. Assuming all three are correct, what is the average of the different possible weights of Arun in kilograms?

Introduction / Context:
This is a logical reasoning question involving ranges and inequalities rather than direct arithmetic only. Several family members estimate Arun's weight with different upper and lower bounds, and we are told that all of them are correct simultaneously. From this we must deduce the common range that satisfies all three statements and then find the average of the possible integer weights in that range.

Given Data / Assumptions:

Arun thinks his weight is greater than 65 kg and less than 72 kg.
His brother thinks Arun's weight is greater than 60 kg and less than 70 kg.
His mother believes Arun's weight is not greater than 68 kg.
All three opinions are correct at the same time.
Weights are assumed to be whole numbers in kilograms for counting different probable values.

Concept / Approach:
Each statement gives an interval of possible values. When all three are correct together, Arun's true weight must lie in the intersection of these three intervals. Once we find this intersection, we list all the different integer values of weight that satisfy it. The question then asks for the average of these different probable weights, which we compute by adding them and dividing by their count.

Step-by-Step Solution:
Step 1: Express each opinion as an inequality range. From Arun: weight is greater than 65 kg and less than 72 kg, so range (65, 72). From brother: weight is greater than 60 kg and less than 70 kg, so range (60, 70). From mother: weight cannot be greater than 68 kg, so weight is less than or equal to 68 kg. Step 2: Find the intersection of these ranges. Intersection of (65, 72) and (60, 70) is (65, 70). Applying the condition from mother (weight ≤ 68 kg) restricts this further to (65, 68]. Step 3: List the different probable integer weights in this intersection. The integer values strictly greater than 65 and less than or equal to 68 are 66 kg, 67 kg and 68 kg. Step 4: Compute the average of these possible weights. Sum of probable weights = 66 + 67 + 68 = 201 kg. Number of probable weights = 3. Average probable weight = 201 / 3 = 67 kg.

Verification / Alternative Check:
We can check that each of the three values 66, 67 and 68 fits all opinions. Each is greater than 65 and less than 72, so Arun's estimate is satisfied. Each is greater than 60 and less than 70, so the brother is also correct. None of them exceeds 68, so the mother is also correct. The average of these three values is clearly 67, confirming our calculation.

Why Other Options Are Wrong:
Option 65 kg is outside the allowed range because Arun claims his weight is greater than 65 kg, not equal to it.
Option 66.5 kg is a reasonable looking decimal but the problem speaks of different probable weights typically interpreted as integer kilogram values, and the exact intersection average of integer possibilities is 67 kg, not 66.5 kg.
Option 68.5 kg exceeds the mother's limit of 68 kg, and so it cannot be correct.

Common Pitfalls:
A frequent mistake is to average the upper and lower bounds of the largest range or to take an average of all boundary numbers from all three people. Another pitfall is to ignore the requirement that all three statements must be simultaneously true, so some students forget to compute the intersection of intervals. Finally, some candidates overlook whether to treat possible weights as integer values and misinterpret the phrase different probable weights.

Final Answer:
The average of the different probable weights of Arun is 67 kg.