In a row of professors, Akash is the first professor from one end. There are 12 professors between Akash and Bhumi. There are 6 professors between Bhumi and Charu. If there are 15 professors standing after Charu in the row, then what is the minimum possible total number of professors in the row?

Introduction / Context:
This reasoning question is closely related to other position based row problems. You have three named professors Akash, Bhumi and Charu, and you know how many professors stand between each pair and how many stand after Charu. The task is to determine the minimum possible total number of professors in the row. These problems test your understanding of positions, distances, and how counts of people between two individuals translate into numeric positions.

Given Data / Assumptions:
- Akash is the first professor in the row, so Akash stands at position 1 from the left. - There are 12 professors between Akash and Bhumi. - There are 6 professors between Bhumi and Charu. - There are 15 professors after Charu in the row. - We are asked for the minimum possible total number of professors in the row.

Concept / Approach:
We convert the description into position numbers. If there are k professors between two people, then their positions differ by k + 1. Akash has a fixed position, so Bhumi gets a fixed position relative to him. Charu can be placed either to the right or to the left of Bhumi while maintaining 6 professors between them. Each choice leads to a different total row length once we account for the 15 professors after Charu. We then choose the configuration with the smallest valid total.

Step-by-Step Solution:
Step 1: Akash is first, so assign Akash position 1. Step 2: There are 12 professors between Akash and Bhumi, so the distance between their positions is 12 + 1 = 13 positions. Thus, Bhumi stands at position 1 + 13 = 14 from the left. Step 3: There are 6 professors between Bhumi and Charu, so their positions differ by 6 + 1 = 7. Step 4: Case 1: Charu is to the right of Bhumi. Then Charu is at position 14 + 7 = 21. If there are 15 professors after Charu, they occupy positions 22 through 36, giving a total of 36 professors in this arrangement. Step 5: Case 2: Charu is to the left of Bhumi. Then Charu is at position 14 - 7 = 7. With 15 professors after Charu, they occupy positions 8 through 22. Thus, the last position in the row is 22, so there are 22 professors in total. Step 6: Both cases satisfy all relative position conditions. The question asks for the minimum possible total, so we select 22 professors from Case 2.

Verification / Alternative check:
Check Case 2 carefully. With Akash at position 1 and Bhumi at position 14, professors at positions 2 through 13 count to 12, which matches the number between Akash and Bhumi. Charu at position 7 leaves positions 8, 9, 10, 11, 12 and 13 between Bhumi at 14 and Charu at 7, a total of 6 professors, as required. From Charu at 7 to the end at position 22, positions 8 through 22 inclusive represent 15 professors, confirming the statement about professors after Charu. Therefore, all conditions hold for a row of length 22.

Why Other Options Are Wrong:
- Totals such as 20 or 21 are too small to allow 15 professors after Charu along with the given distances between Akash, Bhumi and Charu. - A total of 24 or higher is possible but not minimal. The question specifically asks for the minimum number, so larger consistent totals are not correct answers. - Among the listed choices, 22 is the smallest value that fits all positional constraints.

Common Pitfalls:
Learners sometimes assume that the person described first must always be on the left and the others to the right, ignoring the possibility of someone standing to the left of another intermediate person. Another common mistake is miscalculating the relation between the number of people between two positions and the difference in their positions. Remember that if there are k people between two individuals, their positions differ by k + 1. Considering both left and right possibilities and applying this rule correctly ensures a correct solution.

Final Answer:
The minimum possible total number of professors in the row is 22.