Question

At Bridgetown University, there are 45 time periods during the week for scheduling classes, Use the Generalized Pigeon-Hole Principle to determine how many rooms (at least) are needed if 780 different classes are to be scheduled in the 45 time slots.

Short answer

At least 18 rooms are needed per time slot.

Step-by-step solution

Understand the Generalized Pigeon-Hole Principle

The Generalized Pigeon-Hole Principle states that if you have more items than containers, and you distribute the items across the containers, then at least one container must hold more than one item. Mathematically, if you have \(n\) items and \(k\) containers, then at least one container will contain at least \(\lceil \frac{n}{k} \rceil\) items.

Identify the Items and Containers

In this scenario, the 'items' are the 780 different classes, and the 'containers' are the 45 time slots available for scheduling these classes. We need to determine how many rooms are needed per time slot to hold all 780 classes.

Set Up the Equation

Using the formula from the principle, we need to calculate \(\lceil \frac{780}{45} \rceil\). This will tell us the minimum number of rooms needed per time slot to accommodate all classes.

Perform the Calculation

Calculate \(\frac{780}{45}\) which equals 17.3333. Since we can't have a fraction of a room, we round up to the nearest whole number, which gives us \(\lceil 17.3333 \rceil = 18\). Thus, at least 18 rooms are needed in each time slot.

Key concepts

Class Scheduling

Class scheduling is a crucial aspect of managing a university's timetable efficiently. It involves organizing various classes into specific time slots throughout the academic year. When we look at Bridgetown University, we see that they have a total of 45 available time periods for classes to take place each week. This means that the university must allocate these slots wisely to fit all classes into the schedule.
Class scheduling isn't just about finding an available spot; it is also about ensuring the alignment with students' and faculty's availability, and preventing class overlap that could hinder students from attending necessary courses.

• Efficient scheduling ensures optimal use of time slots and resources.
• A well-organized schedule helps maintain a smooth educational operation.
• Proper class distribution avoids conflicts and maximizes participation.

With the Generalized Pigeon-Hole Principle, we are equipped with a method to calculate how to distribute classes over these slots optimally.

Room Allocation

Allocating rooms for scheduled classes involves making sure there are enough physical locations to accommodate all the scheduled activities. This can be particularly challenging when the number of classes exceeds the available rooms, such as in the case of Bridgetown University with its 780 classes.
In room allocation, careful consideration must be given to class size, and proximity to related facilities, along with other logistical factors that might affect attendance and engagement.

• Ensures that each class has a suitable space.
• Prevents overbooking and maximizes room usage.
• Presents solutions in balancing room demand across different times.

Through the Generalized Pigeon-Hole Principle, we determine the need for 18 rooms per time slot, ensuring all classes find a place without chaos.

Mathematical Calculation

At the heart of solving class scheduling challenges is mathematical calculation. By utilizing the Generalized Pigeon-Hole Principle, we simplify a potentially complex allocation process into a manageable formula. The principle states that if you have more classes (items) than time slots (containers), then at least one slot will have multiple classes.
This translates into the need for additional rooms.
The specific calculation here involves dividing the total number of classes by the number of time slots, resulting in a crucial figure:

Calculate \(\frac{780}{45}\)\ resulting in approximately 17.3333.

Use the ceiling function \(\lceil 17.3333 \rceil = 18\) to ensure there's enough space for every class.

This process not only provides a logical basis for room allocation but also shows the power of mathematical principles in real-world scenarios, ensuring effective use of university resources.