Dividing Apples Among People: A Comprehensive Guide

Dividing 10 apples among 2 or 3 people can be a fascinating problem to explore, especially for those interested in combinatorial methods. In this article, we'll delve into the various ways to solve this problem, focusing on a brute force approach using the J programming language. By the end, you'll understand the multiple possibilities and the underlying mathematical principles.

Dividing 10 Apples Among Two People

When dividing 10 apples among two people, the first person can choose any number of apples from 0 to 10. This means there are 11 possible distributions:

Person 1 takes 0 apples, leaving person 2 to take 10 apples. Person 1 takes 1 apple, leaving person 2 to take 9 apples. Person 1 takes 2 apples, leaving person 2 to take 8 apples. Person 1 takes 3 apples, leaving person 2 to take 7 apples. Person 1 takes 4 apples, leaving person 2 to take 6 apples. Person 1 takes 5 apples, leaving person 2 to take 5 apples. Person 1 takes 6 apples, leaving person 2 to take 4 apples. Person 1 takes 7 apples, leaving person 2 to take 3 apples. Person 1 takes 8 apples, leaving person 2 to take 2 apples. Person 1 takes 9 apples, leaving person 2 to take 1 apple. Person 1 takes 10 apples, leaving person 2 to take 0 apples.

Dividing 10 Apples Among Three People

To divide 10 apples among three people, the brute force method reveals that there are 66 possible ways. Let's explore this step-by-step:

Step-by-Step Explanation

1. **First Person's Choices:** The first person can have any number of apples from 0 to 10, which gives 11 possibilities.

2. **Second Person's Choices:** For each choice of the first person, the second person can have any number of apples from 0 to (10 minus what the first person took), which again gives 11 possibilities. This results in a total of 11 x 11 121 combinations.

3. **Third Person's Choices:** For every combination of the first and second person's choices, the third person's number of apples is uniquely determined as 10 minus the sum of the first and second person's apples. This means the third person's choice is also fixed once the first and second person's choices are determined.

4. **Total Combinations:** Therefore, the total number of ways to divide the apples is 11 x 11 x 1 121, but since the third person's distribution is automatically determined, the unique ways are 66.

Brute Force Solution Using J Programming Language

Let's demonstrate this using the J programming language:

{m.n~10/1111  them:{m|0 10|1 9|2 8|3 7|4 6|5 5|6 4|7 3|8 2|9 1|10 0|

For three people:

{m.n~10/6666  them:----------@@@@@@@@@@

Conclusion

Understanding the various ways to divide 10 apples among people showcases the power of combinatorial methods and the J programming language in solving complex problems. Whether you're a mathematician, a programmer, or just curious about such problems, these insights offer a solid foundation for exploring more complex scenarios.