Question

A committee of 5 persons is to be formed from 7 men and 3 women. What is the probability that the committee contains 3 men? (1) \(\frac{5}{36}\) (2) \(\frac{7}{12}\) (3) \(\frac{5}{12}\) (4) \(\frac{1}{3}\)

Short answer

Solution: 1. Calculate total committees possible: \(C(10, 5) = 252\) 2. Calculate committees with 3 men and 2 women: \(C(7, 3) \times C(3, 2) = 105\) 3. Probability: \(\frac{5}{12}\) Answer: The probability of selecting a committee with 3 men and 2 women is \(\frac{5}{12}\).

Step-by-step solution

Find total combinations of forming the committee

First, find the total number of combinations to form a committee of 5 persons from the given 7 men and 3 women. The total number of people is 10, so we can use the combination formula, which is given as follows: \(C(n, r) = \frac{n!}{r!(n-r)!}\) where \(n\) is the total number of items, \(r\) is the number of items to be selected, and \(C(n, r)\) is the number of possible combinations. By substituting \(n = 10\) and \(r = 5\), we get: \(C(10, 5) = \frac{10!}{5!(10-5)!} = 252\) So, there are 252 possible ways to form a committee of 5 persons from 7 men and 3 women.

Find combinations of forming a committee with 3 men

Now, we need to find the number of ways to form a committee containing 3 men and 2 women. To do this, we need to find the number of ways to pick 3 men from the 7 men, and the number of ways to pick 2 women from the 3 women. Then, we'll multiply these two numbers to get the total number of ways to form a committee with 3 men and 2 women. The number of ways to pick 3 men from 7 men is: \(C(7, 3) = \frac{7!}{3!(7-3)!} = 35\) The number of ways to pick 2 women from 3 women is: \(C(3,2) = \frac{3!}{2!(3-2)!} = 3\) Now, multiply these two numbers to find the total number of ways to form a committee with 3 men and 2 women: \(35 \times 3 = 105\)

Find the probability

Now, we can find the probability of forming a committee with 3 men. The probability is the ratio of the number of ways to form a committee with 3 men to the total number of possible committees: \(P = \frac{\text{number of ways to form a committee with 3 men}}{\text{total number possible committees}}\) \(P = \frac{105}{252} = \frac{5}{12}\) The probability that the committee contains 3 men is \(\frac{5}{12}\). So, the correct answer is option (3).

Key concepts

Combination Formula

The combination formula is a crucial part of combinatorics. It helps us calculate how many ways we can choose a set number of items from a larger group, without considering the order of selection. This is particularly useful in scenarios like forming a committee or choosing a team.
The formula is expressed as:

• \(C(n, r) = \frac{n!}{r!(n-r)!}\)

Here, \(n\) represents the total number of items to choose from, while \(r\) is the number of items to be chosen. The symbol \(n!\) denotes the factorial of \(n\), which is the product of all positive integers up to \(n\). The combination formula helps us understand how we can group items in a multitude of ways without worrying about their sequence.
For instance, in the given problem, we have 10 people from which we need to form a committee of 5. So, substituting into the formula gives us \(C(10, 5)\), which calculates the total possible combinations as 252.

Committee Formation

The concept of committee formation using combinations is a classic problem in discrete mathematics. It involves creating a subset from a larger group according to certain criteria.
In the context of the exercise, we're forming a committee of 5 people from a pool of 7 men and 3 women. The exercise specifies that exactly 3 men should be in the committee. This requires us to apply our understanding of combinations twice: once to select the men and once more for the women.

• First, choose 3 men out of the 7 available using the combination formula. This gives us \(C(7, 3)\), which equals 35.
• Next, select 2 women from the 3 available women, calculated as \(C(3, 2)\), giving 3.

By multiplying these two results (35 and 3), we find there are 105 possible committees that meet the criteria of having 3 men and 2 women. This showcases how combinations work hand in hand to solve problems involving specific selection criteria.

Discrete Mathematics

Discrete mathematics is a branch of mathematics that deals with distinct and separated values. It often involves counting and evaluating finite systems, which is vital for tasks such as probability, algorithms, and combinatorics.
In this problem, discrete mathematics helps us systematically analyze the selection of committee members. We're not looking at infinite possibilities. Instead, we have a finite group of people and specific rules about how they can be selected (like forming a committee of exactly 5 members with set gender criteria).
The use of formulas like the combination formula plays a central role in stopping us from manually counting every possible valid group, thereby saving time and effort while eliminating errors. This demonstrates how discrete math provides the tools necessary for logical problem solving in real-world applications.

Probability Calculation

Probability calculation allows us to determine the likelihood of a particular event occurring among possible outcomes. It is often expressed as a fraction or percentage.
To find the probability of forming a committee with 3 men, we divide the number of favorable outcomes by the total possible outcomes. In this problem:

• Total possible ways to form any committee of 5 from 10 people is calculated as 252.
• There are 105 favorable ways to form a committee with exactly 3 men and 2 women, as calculated in the previous sections.

Thus, the probability is \(P = \frac{105}{252}\). Simplifying this fraction gives \(\frac{5}{12}\), indicating the chance that a committee randomly formed will have exactly 3 men.
Understanding probability through combinatorics helps in predicting outcomes and making informed decisions based on data analysis.