Forming a 5-Member Committee from 9 People: A Detailed Guide

When
forming a 5-member committee from a pool of 9 knowledgeable individuals, understanding the principles of combinatorial mathematics is crucial. This article will provide a step-by-step explanation and explore the different methods to calculate the number of possible committees, highlighting the practical applications in various scenarios.

Mathematical Foundation

In combinatorial mathematics, forming a group of k members from a set of n individuals can be represented as C(n, k) or nCk. The formula to calculate this is given by:

C(n, k) n! / (k!(n - k)!)

Calculating the Number of Ways to Form a 5-Member Committee

Let's break down the process of forming a 5-member committee from a group of 9 people:

Method 1: Selecting Members

To form the committee, we can select members one by one, considering the order:

1. For the first member, there are 9 choices.

2. For the second member, there are 8 remaining choices.

3. For the third member, there are 7 remaining choices.

4. For the fourth member, there are 6 remaining choices.

5. For the fifth member, there are 5 remaining choices.

Thus, the total number of ways is:

9 x 8 x 7 x 6 x 5 15,120 ways.

However, since the order of selection does not matter, we must account for the permutations of the 5 members. The number of permutations of 5 members is:

5! 5 x 4 x 3 x 2 x 1 120 ways.

Therefore, the total number of unique committees is:

15,120 / 120 126 ways.

Method 2: Excluding Members

Alternatively, we can consider the number of ways to exclude members:

1. For the first excluded member, there are 9 choices.

2. For the second excluded member, there are 8 remaining choices.

3. For the third excluded member, there are 7 remaining choices.

4. For the fourth excluded member, there are 6 remaining choices.

5. For the fifth excluded member, there are 5 remaining choices.

Thus, the total number of ways is:

9 x 8 x 7 x 6 x 5 15,120 ways.

Again, since the order of excluding members does not matter, we must account for the permutations of the 4 excluded members. The number of permutations of 4 members is:

4! 4 x 3 x 2 x 1 24 ways.

Therefore, the total number of unique committees is:

15,120 / 24 126 ways.

Additional Considerations

Forming a committee not only involves mathematical calculations but also ensures that the selected members possess the necessary qualifications, competence, knowledgeability, and ethical standards. This additional step:

Ensures the quality of the committee is maintained. Avoids the inclusion of unqualified or unethical members. Guarantees the committee represents a diverse range of expertise and experience.

For example, if we have 10 people and want to select a 5-member committee, the number of ways is:

C(10, 5) 10! / (5!5!) 678910 / 2345 479 252 ways.

This can also be read off from Row 10, Position 5 of Pascal’s Triangle.

Conclusion

The total number of ways to form a 5-member committee from 9 people is 126. This understanding is fundamental in various real-world applications, such as project management, team-building exercises, and organizational structures. Proper selection methods ensure the committee's effectiveness and efficiency.

For further reading, consider exploring additional combinatorial scenarios and their practical applications.