Problem 15 - Algebra for College Students

Question

How many three-person committees can be formed from a group of nine people? 84

Step-by-Step Solution

Answer

84 committees can be formed.

Step 1: Identify the Problem Type

This is a combinatorics problem where we need to find out how many ways we can choose a subset of 3 people from a total of 9 people. Since the order of selection doesn't matter, this is a combination problem.

Step 2: Use the Combination Formula

To find the number of ways to choose a subset of items where order doesn't matter, we use the combination formula: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]where \( n \) is the total number of items to choose from and \( r \) is the number of items to choose. In this problem, \( n = 9 \) and \( r = 3 \).

Step 3: Plug Values into the Formula

Substitute the values for \( n \) and \( r \) into the combination formula:\[ C(9, 3) = \frac{9!}{3!(9-3)!} \]which simplifies to:\[ \frac{9!}{3!6!} \].

Step 4: Calculate the Factorials

Calculate each factorial:- \( 9! = 9 \times 8 \times 7 \times 6 \times \dots \times 1 \)- \( 3! = 3 \times 2 \times 1 = 6 \)- \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \).

Step 5: Simplify the Expression

Substitute the factorials back into the equation:\[ \frac{9 \times 8 \times 7 \times 6!}{3! \times 6!} \].The \( 6! \) terms cancel out, leaving:\[ \frac{9 \times 8 \times 7}{3!} \].

Step 6: Finish the Calculation

Explicitly calculate the remaining expression:\[ \frac{9 \times 8 \times 7}{6} = \frac{504}{6} = 84 \].

Step 7: Conclusion

The calculation shows there are 84 different three-person committees that can be formed from a group of nine people.

Key Concepts

Combination Formula

In the world of combinatorics, the combination formula is a powerful tool that helps us determine how many ways we can choose a subset of items from a larger set, without considering the order. This is particularly useful in problems where the order of selection doesn't impact the outcome, such as forming committees or teams.

The combination formula is given by:

• \[ C(n, r) = \frac{n!}{r!(n-r)!} \]

Here:

• \( n \) represents the total number of items or people in the group.
• \( r \) is the number of items or people to choose.
• The exclamation mark \(!\) denotes a factorial, which we will learn more about shortly.

The main takeaway is that combinations focus solely on the selection of items, not their arrangement. In the context of our problem, using combinations helps us find out how many ways we can form a three-person committee from a group of nine people, without worrying about who's chosen first or second.

Factorials

Factorials play a crucial role in the calculation of combinations. A factorial of a number \( n \), represented as \( n! \), is the product of all positive integers up to \( n \). For example, the factorial of 4, written as \( 4! \), is equal to \( 4 \times 3 \times 2 \times 1 = 24 \).

Factorials seem complex at first glance, but they simplify combinations significantly. Their recursive nature means each factorial can be broken down:

• \( n! = n \times (n-1)! \)
• \( 0! \) is defined as 1 for simplifying combinations involving zero selections.

In our three-person committee problem, we use factorials to manage the number of ways items can be arranged and then adjust it to account for the subset size of interest.Calculating factorials:

• 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880
• 3! = 3 \times 2 \times 1 = 6
• 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720

Factorials help reduce the combination formula make it easier to handle by canceling the common factorial terms within the numerator and the denominator.

Subset Selection

Subset selection is what differentiates combinations from permutations. In subset selection through combinations, the focus is on choosing a specific number of elements (the subset) from a larger group, without regard to order. The idea is to capture those unique groupings or collections, such as when forming committees or selecting teams.

When solving problems about forming different types of subsets from a group:

• Identify the total count of items, which is our potential selection pool.
• Decide on the number of items to be selected for our subset, based on the problem's requirements.
• Determine whether the order in which you select matters; if it doesn't, use the combination formula.

In our exercise of forming a three-person committee from nine people:

• The group of nine forms the total set \( n \), where we select a smaller group of three \( r \) for the committee.
• The combination formula doesn't consider arrangements, just the unique groupings possible.

Therefore, through the combination process, you can efficiently determine all possible ways to make subsets, such as committees, while relying on mathematical principles to back those selections.