A store owner packs identical rectangular books into a larger carton that measures 25 inches by 42 inches by 60 inches internally. Each book measures 7 inches by 6 inches by 5 inches. What is the maximum number of books that can be placed in the carton?

Introduction / Context:
Packing optimization problems are common in aptitude tests and practical logistics. Here, we have a large rectangular carton and smaller rectangular books. Because the books fit exactly in integer counts along each dimension, the maximum number of books can be found by seeing how many fit along the length, breadth and height, and then multiplying these counts.

Given Data / Assumptions:

• Carton dimensions: 25 in by 42 in by 60 in.
• Book dimensions: 7 in by 6 in by 5 in.
• Books are identical and rigid.
• Books are oriented with edges parallel to the carton edges, but we can choose the orientation that gives the maximum count.
• Books cannot overlap or be bent.

Concept / Approach:
We must consider all possible orientations of the book in the carton, since placing different edges along the carton dimensions can change how many books fit. For each orientation, we compute:
Number along length = floor(L_carton / L_book). Number along breadth = floor(B_carton / B_book). Number along height = floor(H_carton / H_book). We then multiply these three counts to get the total books for that orientation and choose the maximum among all orientations.

Step-by-Step Solution:
Let carton dimensions be 25, 42, 60. Book dimensions are 7, 6, 5. Consider several orientations: Orientation 1: 7 along 25, 6 along 42, 5 along 60. Counts: floor(25 / 7) = 3, floor(42 / 6) = 7, floor(60 / 5) = 12. Total = 3 * 7 * 12 = 252. Orientation 2: 5 along 25, 7 along 42, 6 along 60. Counts: floor(25 / 5) = 5, floor(42 / 7) = 6, floor(60 / 6) = 10. Total = 5 * 6 * 10 = 300. Other orientations give totals such as 240, 256, 280 or 288, all less than 300. Therefore, the maximum possible number of books is 300.

Verification / Alternative check:
Consider Orientation 2 in more detail. Along the 25 in side, 5 books of thickness 5 in fit exactly. Along the 42 in side, 6 books of width 7 in fit exactly. Along the 60 in side, 10 books of height 6 in fit exactly. No further books can be added in any direction. This packing is efficient and uses the space well, giving confidence that 300 is truly the maximum.

Why Other Options Are Wrong:
175, 225 and 265 books are all less than the achievable 300 and correspond to suboptimal orientations or incomplete use of space.
345 books would require more volume than the carton has and would not fit given the integer constraints on counts along each dimension.

Common Pitfalls:
A common mistake is to calculate only one orientation and assume it is optimal without checking others. Another error is to divide by dimensions incorrectly or to forget to take the floor (integer part) of each division, accidentally assuming fractional books are possible. Systematically checking different orientations and always using whole number counts ensures an accurate answer.

Final Answer:
The maximum number of books that can be packed into the carton is 300 books.