Final answer:
The correct answer is (d) ⁷C2, which represents the number of ways 5 remaining distinguishable apples can be distributed among 3 boys after each has already received one apple.
Step-by-step explanation:
The question asks about the number of ways in which 8 distinguishable apples can be distributed among 3 boys with each boy getting at least one apple. To solve this problem, we use the principle of combinatorics.
Firstly, we need to guarantee that each boy gets at least one apple. We can give one apple to each boy, leaving us with 5 apples to distribute freely among the 3 boys. This problem can be visualized as a stars and bars problem, or more formally, as an application of the formula for combinations with repetition:
- n+k-1 choose k, where n is the number of items to distribute (the 5 remaining apples) and k is the number of categories (the 3 boys).
- The correct formula is (n+k-1)C(k-1), which in this case translates to (5+3-1)C(3-1), or ⁷C2.
This means the correct answer is (d) ⁷C2, representing the number of ways to distribute the remaining 5 apples among the 3 boys after each has received one.