In how many ways can 12 different balls be divided between 2 boys, one receiving 5 and the other 7 balls? Also, in how many ways can these 12 balls be divided into groups of 5,4 and 3 balls, respectively?

Explanation

To solve the given problem, we will break it down into two parts as specified in the question.

### Part 1: Dividing 12 Different Balls Between 2 Boys

1. **Understanding the Distribution**: We have 12 different balls, and we need to divide them between two boys, where one boy receives 5 balls and the other receives 7 balls.

2. **Choosing Balls for the First Boy**: We can choose 5 balls out of 12 for the first boy. The number of ways to choose 5 balls from 12 is given by the combination formula:
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