Forming Teams with Specific Conditions Using Combinatorics

In this article, we will solve a specific problem in combinatorics: forming three 3-person teams from 3 women and 6 men, with the condition that one woman must be on each team. This problem is an excellent example of applying the principles of permutations and combinations to real-world scenarios.

Understanding the Problem

Given 3 women and 6 men, we need to form three 3-person teams such that each team has exactly one woman and two men. Let's break this down step by step:

Step 1: Assigning Women to Teams

First, we need to ensure that each of the three teams has one woman. This can be achieved by selecting one woman for each of the three teams. Since we have 3 women, we can assign one to each team in 3! ways, which equals 6 ways.

Step 2: Distributing Men Among Teams

In the second step, we distribute the 6 men among the three teams, with each team receiving exactly two men.

The number of ways to distribute 6 men into three groups (2 men per team) can be calculated using the multinomial coefficient:

(frac{6!}{2!2!2!})

Calculating this, we get:

Step 3: Combining the Results

To find the total number of ways to form the teams, we multiply the number of ways to assign the women by the number of ways to distribute the men:

3! * (frac{6!}{2!2!2!}) 6 * 90 540

Hence, the total number of ways to split 3 women and 6 men into three 3-person teams with one woman on each team is (540).

Visualizing the Teams

Let's list out the possible combinations of 3-person teams where one woman is in each team:

1. (1W, 2M) from among (3W, 6M)

2. (1W, 2M) from among (1W, 6M)

3. (1W, 2M) from among (1W, 4M)

4. (1W, 2M) from among (1W, 2M, 1W)

5. (1W, 2M) from among (1W, 2M) which respectively (n1(frac{6!}{4!2!}) 15) combinations, 1 (1(frac{4!}{2!2!}) 6) 1 (1(frac{2!}{0!2!}) 1) Hence, the total number of combinations is (15 6 1 22).

Each woman and each man appearing exactly once among the 3-person teams would result in 15 possible 3-person teams.

Conclusion

Understanding and applying combinatorial principles is essential in solving complex problems in a structured manner. The steps outlined here demonstrate how the principles of permutations and combinations can be used to solve real-world problems such as team formation.