Forming Committees with a Majority from Two Party Members

When forming a committee with specific majority requirements, the task involves combinatorial analysis to determine the number of possible ways to achieve the desired outcome. This is particularly relevant in contexts where representation and power dynamics are crucial, such as in political party organizations.

Let's delve into the problem: Given that there are 5 members belonging to party A and 6 members belonging to party B, in how many ways can a committee of 8 members be formed so that party B always has a majority?

Definition of Majority

First, we need to clarify the concept of majority. In most cases, a majority means that a certain group must have more than 50% of the total members. For an 8-member committee, a party must have at least 5 members to have a majority. Thus, we need to determine the number of ways to form a committee where party B has either 5 or 6 members, as this ensures an overall majority.

Combinatorial Analysis

We will analyze the committee by considering two scenarios:

The committee has 5 members from party B and 3 members from party A. The committee has 6 members from party B and 2 members from party A.

Scenario 1: 5 Members from Party B and 3 Members from Party A

The number of ways to choose 5 members from party B (which has 6 members) can be calculated using the combination formula ( C(n, k) frac{n!}{k!(n-k)!} ): [ C(6, 5) frac{6!}{5!(6-5)!} 6 ] Similarly, the number of ways to choose 3 members from party A (which has 5 members) is: [ C(5, 3) frac{5!}{3!(5-3)!} 10 ] Therefore, the total number of ways to form the committee in this scenario is: [ 6 times 10 60 ]

Scenario 2: 6 Members from Party B and 2 Members from Party A

Here, the calculation is as follows:

[ C(6, 6) frac{6!}{6!(6-6)!} 1 ] And for party A: [ C(5, 2) frac{5!}{2!(5-2)!} 10 ] Thus, the total number of ways to form the committee in this scenario is: [ 1 times 10 10 ]

Total Number of Ways

Adding the results from both scenarios, we get the total number of ways to form the committee where party B has a majority:

[ 60 10 70 ]

Conclusion

In conclusion, there are 70 distinct ways to form an 8-member committee from 5 members of party A and 6 members of party B, ensuring that party B has a majority. This combinatorial approach provides a solid foundation for understanding and solving similar problems involving representation and majority dynamics in committees or other similar group formations.

Keywords

majority, combinatorics, committee formation