Distributing Identical Pencils to Students: A Comprehensive Guide

Suppose you have 10 identical pencils and 4 students who need to receive pencils. Each student must get at least one pencil. How can you distribute these pencils effectively?

Step-by-Step Guide to Distributing Pencils

This problem can be solved using combinatorial mathematics, specifically the stars and bars theorem. The theorem provides a method to determine the number of ways to distribute n identical items among k distinct groups.

Step 1: Ensure Each Student Gets at Least One Pencil

Since each student must receive at least one pencil, we start by giving one pencil to each student. This uses up 4 pencils, leaving us with 6 pencils to distribute among the 4 students.

Step 2: Distributing the Remaining Pencils

The problem now is to distribute 6 pencils among 4 students with no restrictions. We use the stars and bars theorem to solve this. The formula is:

Number of ways ( binom{n k-1}{k-1} )

Here, n 6 (remaining pencils) and k 4 (students). Plugging in the values, we get:

Number of ways ( binom{6 4-1}{4-1} binom{9}{3} )

Step 3: Calculate the Binomial Coefficient

The binomial coefficient ( binom{9}{3} ) is calculated as:

[ binom{9}{3} frac{9 times 8 times 7}{3 times 2 times 1} frac{504}{6} 84 ]

Conclusion

The total number of ways to distribute 10 identical pencils to 4 students, ensuring that each student receives at least one pencil, is 84.

Pencil Distribution Insight: The Toblerone Theorem

This method can be visualized using a segmented candy bar like a Toblerone. If you have 10 segments and want to break it into 7 pieces, you can think of the segments as stars and the spaces between them as bars. The number of ways to do this is analogous to the problem of distributing pencils.

Further Exploration: General Case

Let's generalize the problem. Consider a scenario where you have 10 identical pencils and 7 children, each needing at least one pencil. Here, we first give 7 pencils one to each child, leaving 3 pencils to be distributed. The applicable distribution formula is:

Number of ways ( binom{n-r 1}{r-1} )

Plugging in the values, we get:

Number of ways ( binom{10-7 1}{7-1} binom{4}{6} )

Note: The correct formula here should be ( binom{3 7-1}{7-1} binom{9}{6} )

The number of ways to distribute 3 pencils among 7 children is calculated as:

[ binom{9}{6} frac{9!}{6! times 3!} 84 ]

Conclusion and Takeaways

By following the steps outlined in this article, you can systematically distribute identical items among distinct groups while ensuring each group meets a minimum requirement. Whether it's distributing pencils to students or breaking a candy bar, the stars and bars theorem provides a practical and efficient solution.

Keywords: pencil distribution, combinatorics, stars and bars theorem