Combinatorics in Committee Selection: Ensuring Gender Representation

When forming a diverse and representative committee, ensuring that it includes individuals from different backgrounds, such as gender, is crucial for a fulfilling and fair representation. This article explores a specific combinatorial problem to illustrate how to select a committee that must include at least one woman from a group of 7 men and 8 women. We will use the principle of complementary counting and the inclusion-exclusion principle to find the solution.

Introduction to the Problem

The problem is as follows: A club consisting of 7 men and 8 women will choose a committee of 5 members. The committee must contain at least one woman. Our goal is to determine the number of ways this can be achieved.

Solution Using Complementary Counting

Complementary counting is an effective method for solving problems by first calculating the number of ways to achieve the opposite outcome and then subtracting it from the total number of possible outcomes. Here, we will calculate the total number of ways to choose a committee of 5 from 15 members, and then subtract the number of all-male committees.

Step 1: Total Number of Ways to Choose a Committee

The total number of members in the club is 15 (7 men 8 women). The number of ways to choose 5 members from 15 is given by the combination formula:

(n
r) (n!/r!(n-r)!)

Plugging in the numbers:

(15
5) (15!/5!(15-5)! (15!/5!10!) (15 times; 14 times; 13 times; 12 times; 11 /
5 times; 4 times; 3 times; 2 times; 1) 3003)

Hence, there are 3003 ways to form a committee of 5 from 15 members.

Step 2: All-Male Committees

To find the number of all-male committees, we need to choose 5 men from the 7 available. The number of ways to do this is:

(7
5) (7! / 5!7-5! 7! / 5!2!) (7 times; 6 /
2 times; 1) 21)

Step 3: Subtracting All-Male Committees from Total Committees

By subtracting the number of all-male committees from the total number of committees, we get the number of committees that include at least one woman:

(total committees) - (all-male committees) 3003 - 21 2982

Conclusion

The number of ways to choose a committee of 5 that contains at least one woman is 2982.

Additional Explorations: Inclusion-Exclusion Principle

Another approach to solving this problem is through the inclusion-exclusion principle. Let's explore this method as well:

Let M be the set of all possible committees of 5 chosen from 7 men and 8 women. Let W be the set of committees of 5 chosen only from the 7 men (i.e., no women). The number of all-male committees is:

(7
5) (7! / 5!7-5! 7! / 5!2!) (7 times; 6 /
2 times; 1) 21)

The number of committees that are not all-male is given by:

(15
5) - (7
5) (15 times; 14 times; 13 times; 12 times; 11 / 5! - 7 times; 6 / 2!) 3003 - 21 2982)

Combinatorial Options

We can also explore the different combinations of men and women that can form the committee:

2 women and 3 men: (8
2) times; (7
3) 28 times; 35 980) 3 women and 2 men: (8
3) times; (7
2) 56 times; 21 1176) 4 women and 1 man: (8
4) times; (7
1) 70 times; 7 490) 5 women and 0 men: (8
5) times; (7
0) 56 times; 1 56)

The total number of ways to select 5 people from 14 (7 men 8 women) is:

(14
5) (14! / 5!9!) 2002

The overall probability can be calculated as:

(980
1176
490
56 / 2002) 2692 / 2002 ≈ 1.345)

However, this is just the probability of including at least one woman in the committee. The number of ways remains 2982 as calculated earlier.

Conclusion

By using both complementary counting and the inclusion-exclusion principle, we have determined that there are 2982 ways to form a committee of 5 that includes at least one woman. This method not only solves the problem but also provides insight into the combinatorial nature of committee selection and gender diversity.