Question

How many ways can a committee of three be chosen from four teams of two with each team consisting of a man and a woman if: (a) All are equally eligible. (b) The committee must consist of two women and one man. (c) A man and a woman from the same team cannot serve on the committee.

Short answer

(a) 56 ways; (b) 24 ways; (c) 32 ways.

Step-by-step solution

Calculate Total Number of People

Each of the four teams has 2 members, one man and one woman. Therefore, the total number of people is calculated as:\[ 4 \text{ teams} \times 2 \text{ people per team} = 8 \text{ people} \].

Calculate Total Ways to Form a Committee (All Eligible)

To find the total number of ways to choose a committee of 3 from the 8 people if all are equally eligible, use combinations:\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \text{ ways} \].

Find Ways to Form a Committee with Two Women and One Man

Choose 2 women from the 4 available women and 1 man from the 4 available men. Calculate the number of ways:Choosing 2 women:\[ \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6 \text{ ways} \].Choosing 1 man:\[ \binom{4}{1} = 4 \text{ ways} \].Total ways:\[ 6 \times 4 = 24 \text{ ways} \].

Find Ways to Form a Committee (No Same Team)

Total ways without restriction is 56 (from step 2). Now, find invalid committees where a man and woman from the same team are chosen.For each team, choosing the couple:\[ 1 \text{ way per team} \].Choose the remaining member from the other teams (6 choices):\[ 4 \text{ teams} \times 6 \text{ choices} = 24 \text{ invalid ways} \].Subtract invalid from total:\[ 56 - 24 = 32 \text{ valid ways} \].

Key concepts

Combination Formula

In combinatorics, a common problem is to determine how many ways you can choose certain elements from a larger set, without regard to the order of selection. This is where the concept of combinations and the combination formula comes into play. The formula for combinations is given as:

• \[\binom{n}{r} = \frac{n!}{r! \cdot (n-r)!} \]
• Here, \(n\) represents the total number of items to choose from.
• \(r\) is the number of items to be selected.
• \(!\) denotes factorial, which is the product of all positive integers up to that number.

For example, in selecting a committee of three members from eight people, we calculated the number of combinations as \(\binom{8}{3}\). This is interpreted as the number of ways to choose 3 members out of 8, and by applying the formula, we calculated it to be 56 ways.
Understanding this formula is crucial in solving many combinatorial problems, where the order of selection is not important.

Committee Selection

When dealing with committee selection, the goal is to choose a subset from a larger group based on given criteria. In our problem, different constraints affect how the committee is chosen:

• Without restrictions: We choose 3 members out of 8 people, calculated using combinations.
• Specific member composition: Such as requiring 2 women and 1 man. This needs separate selection among women and men followed by multiplication of the separate counts, since selection events are independent.
• Restrictive conditions: Such as avoiding choosing both members from the same team, where we first identify and subtract invalid choices from the total possibilities.

For more complex scenarios, breaking down the steps as seen above helps make the problem more manageable. First, setting up the problem with the desired outcome, applying combination calculations, and finally adjusting for any additional conditions provides a structured approach to solution finding.

Binomial Coefficient

The binomial coefficient, expressed as \(\binom{n}{r}\), is central to the field of combinatorics. It not only helps in counting combinations but also has wide applications in probability and statistics.

• It represents the number of ways to choose \(r\) members out of a total \(n\) without considering the order.
• This coefficient can be calculated using the factorial formula mentioned under the combination formula.

In our example, each step of the committee selection makes use of binomial coefficients:

• Calculating the total number of unrestricted committees from 8 people involved \(\binom{8}{3}\).
• Determining how to select 2 women from a group of 4 through \(\binom{4}{2}\).
• Finally, adjusting for conditions like disqualifying same-team members where subtraction using binomial results helps.

Understanding binomial coefficients equips you with a powerful tool in not just combinatorics but in solving problems across various mathematical applications.