Problem 37 - Calculus for Biology and Medicine

Question

In how many ways can you form a committee of three people from a group of seven if two of the people do not want to serve together?

Step-by-Step Solution

Answer

30 ways.

Step 1: Calculate Total Combinations

First, calculate the total number of ways to choose 3 people out of 7 without any restriction using combinations.The formula for combinations is \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. Here, \( n = 7 \) and \( r = 3 \).\[ \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

Step 2: Identify Restricted Pair

We know that two people do not want to serve together. Let's call them Person A and Person B. To calculate the restricted cases, find the number of committees including both Person A and Person B, and exclude them from the total.

Step 3: Calculate Combinations Including Restricted Pair

Consider the committee where both Person A and Person B are included. With A and B already selected, choose 1 more person from the remaining 5 people.We use the combinations formula \( \binom{n}{r} \), where now \( n = 5 \) and \( r = 1 \).\[ \binom{5}{1} = 5 \]

Step 4: Calculate Valid Committee Combinations

Subtract the restricted cases found in Step 3 from the total combinations found in Step 1 to find how many committees can satisfy all conditions.\[ 35 - 5 = 30 \]

Step 5: Conclusion

The final number of possible committees, adhering to all restrictions, is the number found in Step 4. Ensure there are no other conditions affecting the choices.

Key Concepts

Understanding Combinations

Combinations are a fundamental concept in combinatorics, a branch of mathematics that deals with counting, arrangement, and combination of elements within a set. When we talk about combinations, we are specifically referring to the selection of items from a larger set where the order of selection does not matter.

The basic formula for finding combinations is denoted as \( \binom{n}{r} \), also known as "n choose r". This formula is used to determine the number of ways to choose \( r \) items from a total of \( n \) items. The formula is given by:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

Here, \( n! \) (n factorial) is the product of all positive integers up to \( n \). The factorial function reduces the countings by accounting for the excess combinations where the order doesn't matter.

You can use combinations in various real-world applications, such as in forming teams, distributing awards, or organizing objects, without caring about the order.

Handling Restricted Combinations

Restricted combinations occur when additional conditions or limitations are placed on the selection process, such as certain items cannot be chosen together. In our problem, two people do not want to serve on the committee together. So, we must exclude these scenarios from our calculations.

To tackle restricted combinations, follow these steps:

• First, calculate the total number of unrestricted combinations.
• Next, identify the specific restriction—here, the restriction is combining the two specific people.
• Determine the number of combinations where the restriction is present.
• Finally, subtract the restricted combinations from the total to get the valid number of combinations.

The key in dealing with restricted combinations is to be methodical in calculating and removing the number of unwanted or restricted cases.

Forming Committees Effectively

Committee formation is a common scenario where combinations are applied. When forming a committee, we often have guidelines or restrictions such as role requirements, specific members' preferences, or maximum/minimum capacities.

In this context, a committee is a smaller subset of a larger group that is selected to perform a specific task or function. The key points to consider during committee formation include:

• Determining the total pool of candidates available for selection.
• Understanding any restrictions or criteria each committee must meet.
• Using combinations to calculate the total number of potential configurations.

Forming committees comes down to a problem of selecting some number of items (people) from the bigger group, while taking restrictions into account through the right counting techniques.

Problem-solving Steps for Combinatorial Problems

Solving combinatorial problems often involves systematic steps to ensure all conditions and restrictions are recognized and appropriately handled. To break down the process for the problem at hand:

• Step 1: Calculate Total Combinations – First find the total possible selections without restrictions using the combination formula.
• Step 2: Identify and Isolate Restrictions – Recognize the restricted pairs or groups and list these conditions.
• Step 3: Calculate Restricted Combinations – Find how many combinations include the restricted elements.
• Step 4: Subtract Restricted Cases – Remove the restricted combinations from the total to find the eligible configurations.
• Step 5: Verify Final Number – Cross-check your results to make sure all restrictions are considered and the solution is robust.

By following these structured problem-solving steps, learners can better manage combinatorial challenges with restrictions, ensuring they arrive at the correct and comprehensive solution.