Problem 5 - Algebra for College Students

Question

In how many ways can a committee of three people be chosen from six people? 20

Step-by-Step Solution

Answer

There are 20 ways to choose a committee of three people from six people.

Step 1: Identify the Total Number of People

We start by identifying the total number of people who are available to be chosen for the committee. In this case, there are 6 people.

Step 2: Determine the Number of People to Choose

Next, determine how many people need to be chosen to form the committee. We need to choose 3 people out of the 6 available.

Step 3: Use the Combination Formula

To find the number of ways to choose 3 people from 6, we use the combination formula: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]Where \( n = 6 \) and \( r = 3 \).

Step 4: Calculate the Factorials

Calculate the factorials for the formula:- \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \)- \( 3! = 3 \times 2 \times 1 = 6 \)- \( (6-3)! = 3! = 6 \)

Step 5: Substitute into the Formula

Substitute the calculated factorials into the combination formula:\[ C(6, 3) = \frac{720}{6 \times 6} = \frac{720}{36} = 20 \]

Step 6: Conclude the Calculation

The calculation shows that there are 20 ways to choose 3 people from a group of 6.

Key Concepts

Factorials

Factorials are a fundamental part of combinatorics and mathematics in general. They are particularly useful in calculating permutations and combinations. The factorial of a number, symbolized by the exclamation mark (!), refers to the product of all positive integers up to that number. For instance, the factorial of 4, written as \( 4! \), is calculated as \( 4 \times 3 \times 2 \times 1 = 24 \).
A factorial sequence multiplies until it reaches 1, as shown below:

• \( 1! = 1 \)
• \( 2! = 2 \times 1 = 2 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)

Factorials grow very quickly with larger numbers, making them vital for problems where the order of arrangement matters. In our example problem, factorials were used in the combination formula to determine how many different ways a group can be selected from a larger pool.
Understanding how to calculate factorials is key for students to grasp more complex mathematical concepts, including permutations, probabilities, and series expansions.

Combination Formula

The combination formula is an essential tool in combinatorics and is used to find the number of ways to choose a subset of items from a larger set. This is particularly useful when the order of selection does not matter. The formula is given by:\[ C(n, r) = \frac{n!}{r!(n-r)!} \]Where:

• \( n \) is the total number of items
• \( r \) is the number of items to choose

In this formula, the numerator \( n! \) calculates all possible arrangements of the items, while the denominator \( r!(n-r)! \) accounts for the repetitions and order, which are not important in combinations.
In the example problem, the combination formula was applied with \( n = 6 \) and \( r = 3 \). Plugging these numbers into the formula gave:\[ C(6, 3) = \frac{6!}{3! \times 3!} \]This worked out to 20 different ways to choose 3 people from 6, illustrating how combinations allow for efficient calculation of such problems without having to list all possibilities.

Committee Selection

The concept of committee selection involves choosing a smaller group from a larger set of people or items, where the order of selection does not matter. This scenario is a classic use case for combinations, as it often arises in organizational and academic contexts.
Imagine you have six employees and you need to form a committee of three members. The key is to understand that the arrangement within the committee is irrelevant; it's the composition of the group that matters. Thus, using a combination rather than a permutation is appropriate.
This process becomes straightforward using the combination formula, as seen in our example problem. By applying the combination formula, you can quickly ascertain that there are 20 unique ways to form such a committee, reflecting various configurations where members can be drawn from the larger group of six.

• The selection process in combinatorics and committee scenarios emphasizes the attributes of the subsets chosen.
• It removes the complexity of considering sequences, making it ideal for practical applications where group makeup is the only factor.

Understanding committee selection through combinations gives students insight into more extensive applications of combinatorics beyond academic exercises.