Question

How many ways can two booksellers divide between themselves 300 copies of one book, 200 copies of another, and 100 copies of a third if neither bookseller is to get all the copies of any one of the books?

Short answer

There are 589,701 ways to divide the books.

Step-by-step solution

Understand the Problem

We need to find how two booksellers can divide 300 copies of the first book, 200 copies of the second, and 100 copies of the third, ensuring neither gets all of any book.

Approach the Problem for Each Book Separately

For each type of book, we'll calculate the number of ways they can be divided between the two booksellers, ensuring one gets at least one copy.

Divide 300 Copies of the First Book

For the first book, it's like choosing how many copies the first bookseller gets. There are 299 options because neither book gets all of them. So the number of ways is 299.

Divide 200 Copies of the Second Book

Similarly, for the second book, it's like choosing how many copies the first bookseller gets from 1 to 199, which gives 199 options.

Divide 100 Copies of the Third Book

For the third book, the selection ranges from 1 to 99 since either bookseller getting all is not allowed. Hence, there are 99 ways here.

Calculate Total Division Combinations

The total number of ways to divide all copies is the product of the ways for individual books. So, multiply the ways: \(299 \times 199 \times 99\).

Final Calculation

Compute the product: \(299 \times 199 \times 99 = 589,701\).

Key concepts

permutations

Permutations refer to the various ways in which a set of objects can be ordered or arranged. In our context with the books, we're not necessarily focused on order but rather on how to distribute them between two booksellers.
Permutations become relevant when different arrangements of the same items lead to distinct outcomes. If we were concerned about the order in which the books are divided, or if different copies were unique, then permutations would be the key focus.
For example, if we had a scenario where each copy had a unique identifier, the way they could "line up" to be distributed would count as permutations. But in our problem, the copies are identical, so we focus more on distribution and division methods. It is important to note that the concept of permutations broadens to different contexts and can include ordered arrangements too.
Understanding permutations helps in discrete tasks like arranging students in a line, or seating guests at a table, where the specific order matters.

combinatorial counting

Combinatorial counting involves determining the number of ways to arrange or select items. In the bookseller problem, we're using combinatorial counting to figure out the number of ways to divide the books so neither bookseller gets all of any one type.
Each book type is handled separately. Here's where combinatorial counting shines:

• For the first book, the possible divisions range from the first bookseller getting 1 to 299 copies, excluding the scenario where one bookseller gets all 300 copies.
• For the second book, the division ranges from 1 to 199 copies for the same reason.
• Lastly, for the third book, the division is from 1 to 99 copies.

The crucial step here is multiplying the possible ways for each book. This simplifies larger counting problems by breaking them into manageable parts and using multiplication across independent selections.

discrete mathematics

Discrete mathematics forms the backbone for studying countable, distinct structures, and is integral to computer science, graph theory, and logic problems.
In this problem, discrete mathematics provides the framework to divide the books among booksellers. Here, the discreteness is evident as we deal with whole copies of books, not fractions.
The task of dividing the books ensures that solutions remain within discrete constraints (neither bookseller receives all books).

• The solution implies dividing sets of related items, a classic discrete math scenario.
• This involves using concepts like the pigeonhole principle and integer partitions.

Discrete mathematics allows us to precisely model and solve this type of problem, ensuring that results are both accurate and adhere to the given restrictions.