Selecting a Committee of 7 with Specific Members Included

In combinatorial mathematics, the selection of a group from a larger set is a common problem in probability and discrete mathematics. One such problem involves selecting a committee of 7 members from a group of 11, with the condition that 3 particular members must be included. This article explains in detail how to solve such problems by breaking down the process into simpler steps and using combinatorial formulas.

Step-by-Step Solution

To solve the problem of selecting a committee of 7 members from 11 with 3 specific members included, we can follow these steps:

We start with a total of 11 members and need to include 3 specific members in the committee. This leaves us with 7 - 3 4 members to select from the remaining 11 - 3 8 members.

The number of ways to select 4 members out of 8 can be calculated using the binomial coefficient, denoted as ^8C_4.

Mathematically, the binomial coefficient is defined as:

^nC_r n! / (r! (n - r)!)

Using the formula ^8C_4 8! / (4! × 8 - 4!), we can simplify the expression as follows:

^8C_4 8 × 7 × 6 × 5 / (4 × 3 × 2 × 1)

Carrying out the multiplication and division, we obtain:

^8C_4 7 × 3 × 5 105

Alternative Approach

Another way to approach the problem is by recognizing that since 3 specific members must be included, we need to select only 4 more members from the remaining 8. This can be expressed as:

No. of ways ^8C_4 8! / (4! × 4!)

Simplifying the expression further:

^8C_4 8 × 7 × 6 × 5 / (4 × 3 × 2 × 1) 70

Combinatorial Calculations

Let's break down the steps further to provide a clearer understanding:

1. Out of the 11 members, after including the 3 specific members, we have:

11 - 3 8 members remaining.

2. We need to select 7 - 3 4 more members from these 8:

Remaining members to select 8 - 4 4.

3. The number of ways to select these 4 members is given by the binomial coefficient:

^8C_4 8! / (4! × 4!)

Which simplifies to:

^8C_4 8 × 7 × 6 × 5 / (4 × 3 × 2 × 1) 70

Conclusion

In conclusion, the number of ways to select a committee of 7 members from 11, with 3 particular members included, is calculated as 70. This process of selecting a subset from a larger set, with specific members included, is a fundamental concept in combinatorial mathematics and can be applied in various real-world scenarios, such as forming committees, teams, or groups with specific criteria.

Understanding and applying combinatorial principles can greatly enhance problem-solving skills in areas such as statistics, probability, and discrete mathematics. Whether you are a student, a teacher, or a professional, mastering these concepts can provide a strong foundation in logical reasoning and mathematical analysis.