Calculating the Total Number of Possible Committees

Oct 13, 2025 by recoveryshake.com 81 views

A Four-Person Committee: Calculating the Probability of All Boys

In this article, we will explore the concept of probability in the context of choosing a four-person committee from a group of eight boys and six girls. The problem requires us to calculate the probability that the committee consists of all boys. We will use the concept of combinations to solve this problem and provide a step-by-step solution.

We are given a group of eight boys and six girls, and we need to choose a four-person committee at random. The problem asks us to find the probability that the committee consists of all boys. To solve this problem, we need to calculate the total number of possible committees and the number of committees that consist of all boys.

Calculating the Total Number of Possible Committees

To calculate the total number of possible committees, we need to use the concept of combinations. The number of ways to choose a four-person committee from a group of 14 students (8 boys and 6 girls) is given by the combination formula:

C(n, k) = n! / (k!(n-k)!)

where n is the total number of students (14), k is the number of students to be chosen (4), and ! denotes the factorial function.

Using this formula, we can calculate the total number of possible committees:

C(14, 4) = 14! / (4!(14-4)!) = 14! / (4!10!) = (14 × 13 × 12 × 11) / (4 × 3 × 2 × 1) = 1001

Therefore, there are 1001 possible committees that can be formed from a group of 14 students.

Calculating the Number of Committees with All Boys

To calculate the number of committees that consist of all boys, we need to choose 4 boys from a group of 8 boys. This can be done using the combination formula:

C(n, k) = n! / (k!(n-k)!)

where n is the total number of boys (8), k is the number of boys to be chosen (4), and ! denotes the factorial function.

Using this formula, we can calculate the number of committees with all boys:

C(8, 4) = 8! / (4!(8-4)!) = 8! / (4!4!) = (8 × 7 × 6 × 5) / (4 × 3 × 2 × 1) = 70

Therefore, there are 70 possible committees that consist of all boys.

Calculating the Probability

Now that we have calculated the total number of possible committees and the number of committees with all boys, we can calculate the probability that the committee consists of all boys. The probability is given by the ratio of the number of committees with all boys to the total number of possible committees:

Probability = Number of committees with all boys / Total number of possible committees = 70 / 1001 = 70/1001

Therefore, the probability that the committee consists of all boys is 70/1001.

In this article, we have calculated the probability that a four-person committee chosen at random from a group of eight boys and six girls consists of all boys. We used the concept of combinations to solve this problem and provided a step-by-step solution. The probability is given by the ratio of the number of committees with all boys to the total number of possible committees, which is 70/1001.

The problem of choosing a committee at random is a classic example of a probability problem. The concept of combinations is used to calculate the total number of possible committees and the number of committees with all boys. This problem can be extended to more complex scenarios, such as choosing a committee with a specific number of boys and girls.

The solution to this problem is given by the ratio of the number of committees with all boys to the total number of possible committees, which is 70/1001.

The answer to this problem is:

A. $\frac{4}{1001}$

However, this is incorrect. The correct answer is:

B. $\frac{15}{1001}$

This is because the problem asks for the probability that the committee consists of all boys, and the number of committees with all boys is 70, not 4.

The problem can be solved using the concept of combinations, and the solution is given by the ratio of the number of committees with all boys to the total number of possible committees. The correct answer is $\frac{15}{1001}$ , not $\frac{4}{1001}$ .
A Four-Person Committee: Q&A

In our previous article, we explored the concept of probability in the context of choosing a four-person committee from a group of eight boys and six girls. We calculated the probability that the committee consists of all boys and found that it is 70/1001. In this article, we will answer some frequently asked questions related to this problem.

Q: What is the total number of possible committees that can be formed from a group of 14 students?

A: The total number of possible committees that can be formed from a group of 14 students is 1001. This can be calculated using the combination formula:

C(n, k) = n! / (k!(n-k)!)

where n is the total number of students (14), k is the number of students to be chosen (4), and ! denotes the factorial function.

Q: How many committees can be formed with all boys?

A: There are 70 possible committees that can be formed with all boys. This can be calculated using the combination formula:

C(n, k) = n! / (k!(n-k)!)

where n is the total number of boys (8), k is the number of boys to be chosen (4), and ! denotes the factorial function.

Q: What is the probability that the committee consists of all boys?

A: The probability that the committee consists of all boys is 70/1001. This can be calculated by dividing the number of committees with all boys (70) by the total number of possible committees (1001).

Q: Can the committee consist of all girls?

A: Yes, the committee can consist of all girls. However, the problem does not ask for the probability of this event. If we were to calculate the probability of the committee consisting of all girls, we would need to calculate the number of committees with all girls and divide it by the total number of possible committees.

Q: How can we calculate the number of committees with all girls?

A: To calculate the number of committees with all girls, we need to choose 4 girls from a group of 6 girls. This can be done using the combination formula:

C(n, k) = n! / (k!(n-k)!)

where n is the total number of girls (6), k is the number of girls to be chosen (4), and ! denotes the factorial function.

Using this formula, we can calculate the number of committees with all girls:

C(6, 4) = 6! / (4!(6-4)!) = 6! / (4!2!) = (6 × 5) / (2 × 1) = 15

Therefore, there are 15 possible committees that consist of all girls.

Q: What is the probability that the committee consists of all girls?

A: The probability that the committee consists of all girls is 15/1001. This can be calculated by dividing the number of committees with all girls (15) by the total number of possible committees (1001).

In this article, we have answered some frequently asked questions related to the problem of choosing a four-person committee from a group of eight boys and six girls. We have calculated the total number of possible committees, the number of committees with all boys, and the number of committees with all girls. We have also calculated the probabilities of the committee consisting of all boys and all girls.

The problem of choosing a committee at random is a classic example of a probability problem. The concept of combinations is used to calculate the total number of possible committees and the number of committees with all boys and all girls. This problem can be extended to more complex scenarios, such as choosing a committee with a specific number of boys and girls.

The solutions to this problem are:

The total number of possible committees is 1001.
The number of committees with all boys is 70.
The probability that the committee consists of all boys is 70/1001.
The number of committees with all girls is 15.
The probability that the committee consists of all girls is 15/1001.

The answers to this problem are:

The total number of possible committees is 1001.
The number of committees with all boys is 70.
The probability that the committee consists of all boys is 70/1001.
The number of committees with all girls is 15.
The probability that the committee consists of all girls is 15/1001.