Problem 14 - Calculus for Biology and Medicine

Question

A committee of 3 people must be chosen from a group of 10. The committee consists of a president, a vice president, and a treasurer. How many committees can be selected?

Step-by-Step Solution

Answer

720

Step 1: Understand the Selection Order

Read the problem carefully and identify the roles that need to be filled in the committee: a president, a vice president, and a treasurer. The order in which these roles are filled matters, meaning this is a problem involving permutations.

Step 2: Count President Choices

There are 10 people to choose from to be the president. Therefore, you have 10 options for the president.

Step 3: Count Vice President Choices

Once the president has been selected, there are 9 remaining people to choose from for the role of vice president.

Step 4: Count Treasurer Choices

After selecting the president and the vice president, there are 8 people left to choose from for the treasurer position.

Step 5: Calculate the Total Number of Committees

Multiply the number of choices for each role to find the total number of committees. Thus, the total number of ways to select the committee is given by \( 10 \times 9 \times 8 \).

Step 6: Perform the Calculation

Calculate the value of \( 10 \times 9 \times 8 = 720 \).

Key Concepts

Exploring Combinatorics

Combinatorics is a fascinating area of mathematics that deals with the counting, arrangement, and combination of objects. It's the backbone of problems involving permutations and combinations, making it essential for understanding how we can form different groups or sequences. In the context of committee selection, combinatorics helps us determine the number of ways we can arrange individuals in a committee where roles are crucial, like in our exercise involving a president, vice president, and treasurer.

There are two main types of counting methods in combinatorics: **permutations** and **combinations**. Permutations apply when the order of arrangement matters, which is the case in this problem, while combinations are used when the order does not matter. Understanding these concepts is vital to solving numerous problems in various fields, from mathematics to computer science.

Approaches in Mathematics Education

Mathematics education aims at providing students with the tools and skills to tackle mathematical problems confidently. Permutations and combinatorics can sometimes seem abstract, but breaking them into more relatable concepts, like committee selection, makes them easier to grasp.

One effective method in mathematics education is using **step-by-step processes**. This approach allows students to follow a logical sequence, fostering understanding and retention. By walking through the steps of first choosing a president, then a vice president, and finally a treasurer, educators can clearly demonstrate how permutations work. This hands-on approach encourages students to see patterns and relationships in mathematical operations, ultimately preparing them for more complex problem-solving scenarios.

• Breaking down problems into smaller, manageable parts assists in comprehension.
• Using real-world examples enhances relatability and interest.
• Practicing with varied examples helps solidify concepts in students' minds.

Techniques for Committee Selection

Committee selection often requires careful thought, especially when specific roles need to be filled. The exercise illustrated here demonstrates the method of calculating possible arrangements when selecting a committee of three with distinct roles. This is classic permutation as each role (president, vice president, and treasurer) must be uniquely filled by different individuals, emphasizing the importance of order.

The process runs as follows:

• First, we select one of the 10 available people to be the president.
• Next, from the remaining people, a vice president is chosen, leaving 9 options.
• Finally, we select a treasurer from the 8 people left.

Multiplying these choices together \(10 \times 9 \times 8\) gives 720 different ways to form this specific committee. This approach not only solves the problem but also highlights how permutations are powerful tools in organizing groups efficiently. Understanding why we multiply these numbers is crucial, as it represents how choices made for one role affect subsequent choices.