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

There are 720 different ways to form the committee.

Step 1: Understand the Problem

We need to select 3 people from a group of 10, assigning each person a specific role: president, vice president, and treasurer. This means the order in which we choose the people matters, as each role is distinct.

Step 2: Determine the Number of Ways to Choose the President

Since there are 10 people to choose from initially, there are 10 different ways to select the president of the committee.

Step 3: Determine the Number of Ways to Choose the Vice President

After choosing the president, 9 people remain to choose from for the vice president's position. Thus, there are 9 possible ways to select the vice president.

Step 4: Determine the Number of Ways to Choose the Treasurer

Once the president and vice president have been chosen, 8 people are left. Thus, there are 8 possible ways to select the treasurer.

Step 5: Calculate the Total Number of Committees

Multiply the number of ways to select each role to find the total number of ways to form the committee. This is calculated as follows: \[ 10 \times 9 \times 8 = 720 \]

Key Concepts

Permutations

Permutations are a basic concept in combinatorics, the area of mathematics focused on counting and arrangement possibilities. A permutation refers to the act of arranging all the members of a set into a specific sequence or order. When dealing with permutations, the order of the elements is very important.
If you're asked to find the number of ways to arrange a set or a subgroup of items, permutations come into play. For instance, when forming the committee of president, vice president, and treasurer from 10 people, each unique arrangement of people in these roles is a permutation.
In our problem, the solution is calculated as a permutation because each person holds a distinct position in the committee. So, it’s not just about which people are chosen but who is chosen for each specific role. Using permutations helps us understand and solve problems where the order of selection is crucial.

Role Assignment

Assigning roles means giving specific positions to selected individuals. This directly connects to permutations because when roles are assigned, the order matters. Imagine if choosing people was all there was to it—then you'd just need combinations. But role assignment is different.
In the committee problem, selecting three people from ten and deciding who becomes president, vice president, and treasurer requires assigning roles. It complicates the selection process because you need to consider not just the people but also who gets each specific title.
The number of role assignments can exponentially increase. For our exercise, after determining the first role, the remaining individuals need to be assigned to the other positions, giving us multiple combinations for each subsequent choice.

Factorial

Factorial notation simplifies the calculation of permutations. The factorial of a number, represented as \( n! \), is the product of all positive integers up to \( n \). It's commonly used in permutations as it helps count the number of ways to arrange a set.

• For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Factorials help when you need the total number of arrangements for distinct items. In cases like choosing people for different roles, factorial is a powerful tool. It's used in the formula for permutations, \( P(n, r) = \frac{n!}{(n-r)!} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to arrange.
In our problem, though we directly calculated by multiplying the options for each role, understanding factorials provides a deeper insight into "how" these calculations can be systematically approached.