Twelve distinct balls are to be split between two distinct boys (Rahul and Rohan) so that one boy receives 5 balls and the other receives 7 balls. In how many distributions is this possible?

Introduction / Context:
We distribute distinct items between two labeled recipients with a fixed split of counts. There are two symmetric cases: Rahul gets 5 (Rohan 7) or Rahul gets 7 (Rohan 5).

Given Data / Assumptions (clarified):

• 12 distinct balls; two distinct boys (Rahul, Rohan).
• One boy must get exactly 5 balls; the other must get 7 (not specified which).

Concept / Approach:
Case-split by who gets 5: choose 5 for Rahul (C(12,5)), or choose 5 for Rohan (another C(12,5)). These cases are disjoint and cover all valid allocations.

Step-by-Step Solution:
Number of 5-ball subsets = C(12,5) = 792.Two choices for who receives 5 ⇒ total = 2 * 792 = 1584.

Verification / Alternative check:
Equivalently, choose which 7 go to the 7-ball recipient: 2*C(12,7) = 2*792 = 1584 (since C(12,5)=C(12,7)).

Why Other Options Are Wrong:
Values near 1600 come from arithmetic slips; 1784/1560/1854 are not equal to 2*C(12,5).

Common Pitfalls:
Answering C(12,5)=792 only (implicitly fixing which boy gets 5) despite the statement allowing either boy to be the 5-ball recipient.

Final Answer:
1584