Problem 18 - A First Course in Probability

Question

A committee of \(7,\) consisting of 2 Republicans, 2 Democrats, and 3 Independents, is to be chosen from a group of 5 Republicans, 6 Democrats, and 4 Independents. How many committees are possible?

Step-by-Step Solution

Answer

There are 600 possible committees that can be formed consisting of 2 Republicans, 2 Democrats, and 3 Independents from the given group of 5 Republicans, 6 Democrats, and 4 Independents.

Step 1: Understand the concept of combinations

A combination is a selection of items from a larger set, where the order of the items does not matter. The number of combinations of choosing 'r' items from a set of 'n' items is denoted as C(n, r), and it is calculated using the formula: \[ C(n, r) = \frac{n!}{r! (n - r)!} \] Where 'n!' represents the factorial of n (n! = n * (n-1) * (n-2) * ... * 1), and 'r' and 'n - r' are the factorials of r and n - r, respectively.

Step 2: Find the combinations for each group

To form the committee, we need to choose 2 Republicans, 2 Democrats, and 3 Independents. We can calculate the number of combinations for each group separately: 1. Republicans: We have 5 Republicans and we need to select 2. The combinations for Republicans can be calculated as: C(5, 2) = \( \frac{5!}{2! (5 - 2)!} \) 2. Democrats: We have 6 Democrats and we need to select 2. The combinations for Democrats can be calculated as: C(6, 2) = \( \frac{6!}{2! (6 - 2)!} \) 3. Independents: We have 4 Independents and we need to select 3. The combinations for Independents can be calculated as: C(4, 3) = \( \frac{4!}{3! (4 - 3)!} \)

Step 3: Calculate the combinations for each group

Now, we can calculate the number of combinations for each group: 1. Republicans: C(5, 2) = \( \frac{5!}{2! (5-2)!} \) = \( \frac{5!}{2! 3!} \) = \( \frac{5 * 4 * 3!}{2! 3!} \) = \( \frac{5 * 4}{2} \) = 10 2. Democrats: C(6, 2) = \( \frac{6!}{2! (6-2)!} \) = \( \frac{6!}{2! 4!} \) = \( \frac{6 * 5 * 4!}{2! 4!} \) = \( \frac{6 * 5}{2} \) = 15 3. Independents: C(4, 3) = \( \frac{4!}{3! (4-3)!} \) = \( \frac{4!}{3! 1! } \) = \( \frac{4 * 3!}{3! } \) = 4

Step 4: Calculate the total number of committees

Now we have the number of combinations for each group. To find the total number of possible committees, we multiply the combinations for each group: Total Committees = Combinations of Republicans * Combinations of Democrats * Combinations of Independents Total Committees = 10 * 15 * 4 = 600 So, there are 600 possible committees that can be formed.

Key Concepts

Factorial Notation

Factorial notation is an essential mathematical concept often represented by an exclamation mark (!). It specifically refers to the product of an integer and all the non-zero integers below it. For instance, the factorial of 5, denoted as 5!, is calculated as:
\[\begin{equation}5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\text{.}\text{\end{equation}\]}
Factorials play a crucial role in various areas of mathematics but are particularly noteworthy in probability and combinatorics. When determining the number of ways in which a set of things can be arranged, factorial notation provides a streamlined approach.

As numbers ascend, their factorial values grow exponentially, which is important to remember since the factorial of zero is defined to be 1 (0! = 1). This might initially seem counterintuitive, but it is pivotal when using the combination formula, as a larger set's subsets will undoubtedly involve the factorial operation to calculate combinations.

Combination Formula

In probability and statistics, the combination formula is a pivotal mathematical tool for determining the number of ways to select items from a larger pool where the order of selection doesn't matter. It's defined as:\[\begin{equation}C(n, r) = \frac{n!}{r!(n-r)!}\text{,}\text{\end{equation}\]}
where C(n, r) represents the number of combinations of r items selected from a set of n items. Here, n! denotes the factorial of n, and r! is the factorial of r. The term (n-r)! is the factorial of the difference between n and r.

For instance, if you wish to choose 2 members from a group of 5 to form a subcommittee, you don't concern yourself with the order in which they're chosen; hence, combinations are used instead of permutations. The formula simplifies the calculation process, allowing for quick and easy computation of complex selection problems.

Example of Combination Formula in Use

To choose 2 Republicans from a group of 5, we apply the combination formula:\[\begin{equation}C(5, 2) = \frac{5!}{2! \cdot (5-2)!} = \frac{5 \times 4}{1 \times 2} = 10\text{.}\text{\end{equation}\]}
This tells us there are 10 ways to form a subcommittee of 2 Republicans from a group of 5.

Probability Theory

Probability theory is the branch of mathematics concerned with the analysis of random phenomena. The core goal is to understand and quantify the likelihood of events occurring. Probability is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

A foundational aspect of probability theory is the study of events and outcomes that stem from a process that cannot be predicted with absolute certainty. This includes simple scenarios like flipping a coin or more complex analyses such as forecasting weather patterns.

Probability theory lays the groundwork for calculating the likelihood of various combinations when the order of the elements does not matter. In our committee selection example, probability theory enables us to determine the odds of forming different committees based on party affiliation, eschewing the sequence in which members are selected. This approach is crucial when dealing with real-world scenarios where the outcome is based on numerous factors, such as political committee formation, biological studies, and others where order is not an overriding factor.