Distributing Apples: A Combinatorial Challenge

How many ways can 25 identical apples be distributed among 7 non-identical students such that no one has more than 5 apples? This intriguing problem in combinatorics is best approached using the stars and bars method, an effective tool in solving such distribution problems.

Stars and Bars: An Overview

The stars and bars concept is a powerful combinatorial technique used to solve distribution problems. Essentially, stars represent the identical items (apples in this case), and bars represent the separation between different groups (students in this problem).

Applying Stars and Bars

First, we recognize that the total number of apples is 25, and we want to ensure that no student receives more than 5 apples. A practical starting point is to give each student 5 apples first. Since 7 students each get 5 apples, that accounts for 35 apples in total. However, we only have 25 apples, so we need to remove 10 apples.

The Distribution Process

We begin by considering the problem of distributing 10 apples among 7 students without any restrictions. This can be modeled using the stars and bars method:

[ binom{10 7 - 1}{7 - 1} binom{16}{6} ] However, this approach overcounts scenarios where one or more students receive more than 5 apples. Specifically, we need to subtract the cases where one or more students receive 6 or more apples.

Subtracting Overcounted Cases

First, let's consider one student receiving 6 apples. This leaves 4 apples to be distributed among the remaining 6 students:

[ 7 times binom{4 6 - 1}{6 - 1} 7 times binom{9}{5} ] Repeating this for all students:

[ 7 times binom{9}{5} ] Subtracting the overcounted cases, we get:

[ binom{16}{6} - 7 times binom{9}{5} 6538 ] Thus, there are 6538 ways to distribute the 25 apples such that no one has more than 5 apples.

Exploring Distribution Possibilities

For a deeper understanding, we can list out the possibilities in decreasing size:

55555 145555 235555 244555 334555 344455 444445 1135555 1144555 1225555 1234555 1244455 1333555 1334455 1344445 1444444 2224555 2233555 2234455 2244445 2333455 2334445 2344444 3333355 3333445

Each of these possibilities represents a unique way to distribute the apples, and the number on the right indicates the number of arrangements for each distribution.

Conclusion

By carefully applying combinatorial techniques and accounting for overcounted cases, we can effectively determine the number of ways to distribute 25 identical apples among 7 non-identical students, ensuring no one receives more than 5 apples. The problem showcases the elegance and utility of combinatorial methods in solving real-world distribution challenges.