Problem 56 - Algebra 2 and Trigonometry

Question

The program director at the local sports center wants to schedule the following classes on Wednesdays: aerobics, tai chi, yoga, modern dance, fitness, and weight training. Starting at \(9 : 00\) A.M. each class begins on the hour and ends at 10 minutes before the hour. a. How many different ways can the classes be scheduled? b. If the yoga instructor is available only for a \(9 : 00\) A.M.class, how many different ways can the classes be scheduled?

Step-by-Step Solution

Answer

a. 720 ways; b. 120 ways.

Step 1: Determine the number of classes

Count the total number of classes that need to be scheduled. These are aerobics, tai chi, yoga, modern dance, fitness, and weight training, making a total of 6 classes.

Step 2: Calculate total schedule permutations

Using the formula for permutations, we can calculate the total number of ways to schedule the classes. This is given by the factorial of the number of classes, which is \(6!\). \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\). So there are 720 different ways to schedule the classes.

Step 3: Constraint application for yoga class

The yoga class must be scheduled at 9:00 A.M. This leaves 5 other classes to schedule in the remaining 5 time slots.

Step 4: Calculate permutations with constraint

Since the yoga class is fixed at 9:00 A.M., only the remaining 5 classes need to be scheduled. The number of permutations for these classes is \(5!\). \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). So there are 120 different ways to schedule the classes with the yoga class at 9:00 A.M.

Key Concepts

Factorial Calculations

Factorial calculations are a fundamental concept when dealing with permutations. A factorial is a function that multiplies a series of descending natural numbers. It is denoted by an exclamation mark, such as in \(6!\), which is read as "six factorial." Here is what you need to know about factorials:

• They represent the total number of ways to arrange a set of items.
• For instance, if you have 6 unique items, the number of ways they can be arranged is \(6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\).

In the context of scheduling classes, calculating \(6!\) gives us the total number of possible schedules for six classes. When calculating, always multiply all integers from your number down to one. Factorials help simplify permutation calculations significantly.

Scheduling Problems

Scheduling problems involve determining the order in which a set of tasks or events occur. These types of problems often require understanding permutations and constraints. Here’s why they can be challenging:

• Each task must fit within a defined time frame without overlap.
• Constraints (like certain instructors being unavailable at particular times) can complicate scheduling.

For scheduling the mentioned classes, we start with 6 distinct classes. Each represents a task that needs a unique time slot. Permutation calculations help to determine all possible orders, leading to efficient scheduling.

Constraint Application

Constraints are specific conditions or limitations that must be adhered to in scheduling or problem-solving. Applying constraints requires adjusting calculations to suit the given conditions. Here, constraints simplify complexity by reducing possibilities:

• Identify the constraint first, such as the yoga class that must occur at 9:00 A.M.
• Lock this class in its required time slot, reducing the total items to consider.

In our problem, once the yoga class is set at 9:00 A.M., we only have 5 classes to schedule in 5 slots. Therefore, instead of computing \(6!\), we compute \(5!\), finding that the remaining classes can be arranged in 120 unique ways. Constraints guide us towards more manageable, focused calculations.