Understanding the Warehouse Storage Problem

This problem asks for the minimum number of products a company needs to have so that, when stored in a warehouse, at least two products are guaranteed to be in the same storage bin. This is a classic application of the Pigeonhole Principle.

Applying the Pigeonhole Principle

The Pigeonhole Principle states that if you have more items than containers, then at least one container must hold more than one item. In this scenario:

• The products are the "pigeons".
• The storage bins are the "pigeonholes".

To guarantee that at least two products are stored in the same bin, the number of products must be one more than the total number of available storage bins.

Calculating the Total Number of Storage Bins

The storage bins are defined by three dimensions:

• Aisle number
• Horizontal location within the aisle
• Shelf number

We are given the number of options for each dimension:

• Number of aisles = 50
• Number of horizontal locations per aisle = 85
• Number of shelves = 5

The total number of unique storage bins is the product of these numbers:

\( \text{Total Bins} = \text{Number of Aisles} \times \text{Number of Horizontal Locations} \times \text{Number of Shelves} \)

Let's calculate this value:

\( \text{Total Bins} = 50 \times 85 \times 5 \)

We can perform the multiplication step-by-step:

\( 50 \times 85 = 4250 \)

\( 4250 \times 5 = 21250 \)

So, there are 21250 distinct storage bins in the warehouse.

Determining the Minimum Number of Products

According to the Pigeonhole Principle, to guarantee that at least two products share a bin, the number of products must be one more than the total number of bins.

\( \text{Minimum Products} = \text{Total Bins} + 1 \)

\( \text{Minimum Products} = 21250 + 1 \)

\( \text{Minimum Products} = 21251 \)

Therefore, the company must have at least 21251 products to ensure that at least two products are stored in the same bin.

Summary of Calculation

This calculation confirms that with 21251 products and 21250 bins, at least one bin must contain more than one product.

Revision Table: Key Concepts

Additional Information: Pigeonhole Principle Examples

The Pigeonhole Principle is a fundamental concept in combinatorics and discrete mathematics. Here are a few simple examples to illustrate its use:

• If you have 13 people, at least two of them must have been born in the same month (13 people, 12 months).
• If you pick 5 cards from a standard deck of 52, at least two cards must be of the same suit (5 cards, 4 suits).
• In any group of 6 people, there are either 3 mutual friends or 3 mutual strangers (Ramsey Number \(R(3,3)=6\), a related concept).

Understanding this principle helps solve problems involving guarantees about distributions when items are placed into categories or containers.