Combinations in Selecting Committee Members: A Detailed Analysis

When determining the number of ways to select committee members, it's essential to carefully consider the parameters provided. In the original problem, the confusion arises from the initial specification of individuals and the final selection criteria.

This article aims to clarify and solve the problem accurately. We will explore the correct solution by selecting 3 gentlemen and 2 ladies from a total pool of 8 gentlemen and 12 ladies. This format ensures that the problem is well-defined and adheres to standard combinatorial principles.

Definition and Explanation

Combinations refer to the number of ways to choose a subset of items from a larger set, where the order of selection does not matter. Mathematically, the number of combinations of n items taken r at a time is given by:

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

Solution Analysis

Let's break down the problem into clear steps:

The number of ways to choose 3 gentlemen from 8 is given by the combination formula:

C(8, 3) 8! / (3!(8-3)!) 8! / (3!5!) 56

The number of ways to choose 2 ladies from 12 is given by the combination formula:

C(12, 2) 12! / (2!(12-2)!) 12! / (2!10!) 66

Since these selections are independent, the total number of ways to form a committee of 3 gentlemen and 2 ladies is the product of the individual combinations:

Total ways C(8, 3) * C(12, 2) 56 * 66 3696

Conclusion

By clearly defining the problem and applying the combination formula appropriately, we can determine the exact number of ways to form a committee from the given individuals. In this case, there are 3696 ways to select a committee consisting of 3 gentlemen and 2 ladies from a total of 8 gentlemen and 12 ladies.

Additional Insights

Understanding combinations is crucial in many areas, including probability, statistics, and real-world applications such as elections and team formations. This detailed analysis provides a clear and systematic approach to solving similar problems.

In conclusion, the correct solution to the problem is 3696 ways. This result ensures that the selection process adheres to the given constraints and provides a definitive answer to the question posed.