Problem 73 - Algebra and Trigonometry

Question

55–75 Solve the problem using the appropriate counting principle(s). Choosing a Committee In how many ways can a committee of four be chosen from a group of ten if two people refuse to serve together on the same committee?

Step-by-Step Solution

Answer

182 ways to form the committee without both individuals together.

Step 1: Understand the Problem

We need to form a committee of four people from a group of ten, where two specific individuals refuse to be on the same committee together. Our task is to determine the number of possible valid committees without both of those individuals on the same committee.

Step 2: Total Ways to Form a Committee

First, calculate the total number of ways to choose 4 people from 10 without any restrictions. This can be done using the combination formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). For \( n = 10 \) and \( r = 4 \), this is \( \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \).

Step 3: Identify the Restricted Scenario

The restriction is that two particular people (say, A and B) do not want to serve together. Therefore, we must calculate how many committees include both A and B.

Step 4: Ways with Restrictions

If both A and B are to be included in the committee, we now choose 2 more members from the remaining 8 people. This number is given by \( \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28 \).

Step 5: Calculate Committees Avoiding Restrictions

Subtract the number of restricted committees from the total number of committees calculated in Step 2: \( 210 - 28 = 182 \).

Step 6: Final Answer

Hence, the number of ways to form a committee of four such that the two specific people are not both in the committee is 182.

Key Concepts

Combination Formula

In mathematics, the combination formula is a fundamental tool used to calculate the number of ways to choose a subset of items from a larger set without regard to the order of selection. This is particularly handy in scenarios where the arrangement of chosen items does not matter, like forming a committee.

The combination formula is expressed as:
\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]where:

• \( n \) is the total number of items to choose from.
• \( r \) is the number of items to choose.
• \( ! \) denotes factorial, which is the product of all positive integers up to that number.

Using this formula allows us to determine the number of different sets of \( r \) items that can be selected from \( n \) items without regard to order.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of objects. It provides the methodology for solving problems where we need to count the different ways in which objects can be grouped.

It is essential when solving problems like committee selection, tournament scheduling, and resource allocation. Key operations include:

• Permutations - where order matters.
• Combinations - where order does not matter.
• Factorials - used in both permutations and combinations.

By applying these principles, we can solve complex mathematical problems efficiently and accurately, such as those involving restrictions or specific conditions.

Committee Selection

Selecting a committee involves choosing a specific number of members from a larger group to fulfill particular roles or represent interests. This is a common application in combinatorics where the order of selection doesn't matter; only the group composition does.

In our example, you need to find how many ways a committee of four can be formed from ten possible members. Initially, without restrictions, we use the combination formula \( \binom{n}{r} \) to find the total combinations. This value is 210 possible ways for 10 people.

It is crucial to consider any constraints, as they will affect the final number of valid committees. Applying these constraints accurately to find a valid number of committees is a key problem-solving aspect of such exercises.

Restricted Scenarios

When forming groups like a committee, restrictions can complicate calculations. A common type of restriction involves conditions where certain members cannot be part of the group together.

In the example provided, two individuals cannot be on the committee simultaneously. To handle such restrictions, we first calculate the number of unrestricted combinations, then calculate the scenarios where the restriction is not met (both individuals being in the same group).

You then subtract the restricted scenarios from the total combinations to find all valid committee configurations.
For instance, we found 28 committees where both restricted individuals were included. Subtracting these from the total gives us 182 valid committee formations. This systematic approach ensures accurate and efficient problem-solving.