The following five three-digit-numbers are given as 234, 452, 346, 762, 893If the positions of the first and the second digits in each of the above numbers are interchanged, which of the following will be the difference between the middle digits of the highest and the lowest of the five numbers ?

The correct answer is (c).

If the positions of the first and the second digits in each of the above numbers are interchanged, the new numbers will be:

• 423
• 524
• 643
• 267
• 389

The middle digits of the highest and the lowest of the five numbers are 6 and 3, respectively. Therefore, the difference between them is $6-3=\boxed{3}$.

Here is a brief explanation of each option:

• Option (a): 3. This is the difference between the middle digits of the second and the third numbers in the original list. However, the second and the third numbers are not the highest and the lowest numbers in the list. Therefore, this cannot be the answer.
• Option (b): 5. This is the difference between the middle digits of the third and the fourth numbers in the original list. However, the third and the fourth numbers are not the highest and the lowest numbers in the list. Therefore, this cannot be the answer.
• Option (c): 6. This is the difference between the middle digits of the highest and the lowest numbers in the list. Therefore, this must be the answer.
• Option (d): 7. This is the difference between the middle digits of the first and the fifth numbers in the original list. However, the first and the fifth numbers are not the highest and the lowest numbers in the list. Therefore, this cannot be the answer.