In a row of boys, there are 6 boys between A and B, and A is the first boy in the row. There are 3 boys between B and C. If there are 12 boys standing after C, then what is the minimum possible number of boys in the row?

Introduction / Context:
This question is about positions in a single row of boys. It describes the relative positions of three boys A, B and C and asks for the minimum number of boys in the entire row. Solving this type of problem requires interpreting the phrases "boys between" correctly and considering different possible arrangements to find the smallest consistent total.

Given Data / Assumptions:

• A is the first boy in the row, so his position from the left is 1.
• There are 6 boys between A and B.
• There are 3 boys between B and C.
• There are 12 boys after C, meaning to the right of C in the row.
• We are asked for the minimum possible total number of boys in the row.
• We assume all positions are filled by exactly one boy and there are no gaps.

Concept / Approach:
First, convert "boys between" into distances in terms of positions. If there are 6 boys between A and B, the difference in their positions is 6 + 1 = 7. Next, compute the possible positions for C relative to B, taking into account that there are 3 boys between B and C. C could be to the right or to the left of B. We must also account for the 12 boys after C and then select the arrangement that yields the smallest total number of boys while satisfying all conditions.

Step-by-Step Solution:
Step 1: Since A is the first boy, A is at position 1 from the left.Step 2: There are 6 boys between A and B, so the distance between their positions is 6 + 1 = 7. Because A is at position 1, B must be at position 1 + 7 = 8.Step 3: There are 3 boys between B and C, so the distance between their positions is 3 + 1 = 4. Thus C can be at position 8 + 4 = 12 to the right of B or at position 8 - 4 = 4 to the left of B.Step 4: Next, there are 12 boys after C. If C is at position 12, then there are 12 boys to the right, so the total length of the row would be 12 (position of C) + 12 = 24.Step 5: If C is instead at position 4, then, with 12 boys after C, the last boy is at position 4 + 12 = 16, so the row has 16 boys.Step 6: Both arrangements are consistent with the conditions, but the question asks for the minimum possible number of boys, so we choose the smaller total of 16.Step 7: Therefore the minimum possible number of boys in the row is 16.

Verification / Alternative check:
We can quickly sketch the arrangement that gives 16 boys. Place A at position 1 and B at position 8, leaving positions 2 to 7 between them, which contains 6 boys, satisfying the first condition. Place C at position 4 so that positions 5, 6 and 7 lie between C and B, giving 3 boys between B and C. Finally, place 12 boys after C, occupying positions 5 through 16. All conditions are satisfied with a total of 16 boys.

Why Other Options Are Wrong:
A total of 20 or 24 boys is possible in some configurations, but 16 is smaller and still satisfies all conditions, so those larger totals are not the minimum. A total of 18 boys would require a different position for C or B that would break the given spacing constraints. Therefore those alternatives do not correspond to the minimum valid arrangement.

Common Pitfalls:
Students may misinterpret "boys between" and simply add or subtract incorrectly, or they may forget to consider both possible sides for C relative to B. Another mistake is to ignore the requirement of minimum total and accept the first valid arrangement rather than checking for a smaller consistent arrangement.

Final Answer:
The minimum possible number of boys in the row is 16.