Question

How many four-person teams can be formed from three men and five women if at least one man and at least one woman are on each team?

Short answer

65 teams can be formed.

Step-by-step solution

Identify the Total Number of People

We have a total of 8 people: 3 men and 5 women. We need to form a team of 4 people, maintaining at least one man and one woman on each team.

Total Ways to Form a Team of 4

Calculate the total number of ways to select any 4 people from the 8 available, regardless of gender. This is done using the combination formula: \[\binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70\]

Cases Violating Conditions

Consider cases where the formed team does not meet the requirement of "at least one man and one woman." These are the cases where the team comprises only men or only women.

Only Men

Calculate the number of ways to form a team with only men. This would mean choosing 4 men from 3, which is not possible. Thus, the number of such cases is:\[\binom{3}{4} = 0\]

Only Women

Calculate the number of ways to form a team with only women. This would mean choosing 4 women from 5:\[\binom{5}{4} = 5\]

Apply the Principle of Complementation

Use the principle of complementation to exclude the invalid cases from the total. That is:\[70 - (0 + 5) = 65\]

Confirm At Least One Man and One Woman

After subtracting the invalid teams, the remaining cases (65) have at least one man and one woman, satisfying the given condition.

Key concepts

Combination Formula

In combinatorics, the combination formula is a fundamental tool that helps us determine the number of ways to choose a specific number of items from a larger set, without regard to the order of selection.
When solving problems involving combinations, it's important to use the combination formula:\[\binom{n}{r} = \frac{n!}{r! \, (n-r)!}\]where:

• \(n\) is the total number of items,
• \(r\) is the number of items to choose,
• \(n!\) denotes "n factorial," which is the product of all positive integers up to \(n\).

This formula allows us to calculate the number of possible combinations efficiently. For instance, in the given exercise, you can determine how many ways to form a team of four from a group of eight by using \(\binom{8}{4}\).
This calculation considers all possible 4-person teams without considering who specifically fills each role.

Principle of Complementation

The principle of complementation is a powerful concept in probability and combinatorics, allowing for easier solutions by considering the opposite of what is required.
Instead of directly counting the valid cases, we often find it simpler to count the complement, or invalid cases, and subtract it from the total.
This method is particularly helpful when the direct calculation of valid outcomes seems cumbersome.In our exercise, the main requirement was to ensure at least one man and one woman in each team. To simplify, we first determined the total number of 4-person teams possible using the combination formula: \(70\).
Then, we evaluated configurations that violated the requirement:

• Teams with only men (0 ways, since there are only 3 men)
• Teams with only women (5 ways, calculated as \(\binom{5}{4}\))

Thus, using the principle of complementation, these restrictive cases (5 invalid teams) are subtracted from the total possible teams (70).
The result is 65 acceptable teams, each having at least one man and one woman.

Discrete Mathematics

Discrete mathematics is a branch of mathematics that deals with discrete elements, as opposed to continuous elements.
It often involves counting methods, structures, and theoretical concepts like graph theory, logic, and combinatorics. These tools are invaluable in computer science, optimization, and routing problems. In the context of forming teams, discrete mathematics helps identify how groups can be constructed from discrete sets of people.
Combinatorics, as showcased in today’s exercise, plays a crucial role in such scenarios.
By using combinations and the principle of complementation, one can systematically manage the arrangement and selection of discrete groups to fulfill specific criteria. Understanding these discrete concepts aids in solving many real-world problems where items are distinct, and arrangements must adhere to set rules or conditions.
Because it focuses on finite or countably infinite structures, discrete mathematics provides clarity and efficiency in both academic and practical applications.