Problem 60 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In how many different ways can a panel of 12 jurors and 2 alternate jurors be chosen from a group of 30 prospective jurors?

Step-by-Step Solution

Answer

There are 5,153,875 different ways to choose a panel of 12 jurors and 2 alternate jurors from a group of 30 prospective jurors. This is calculated using combinations: total ways = C(30, 12) × C(18, 2) = \(\frac{30!}{12!} × \frac{1}{16!2!}\) = 5,153,875.

Step 1: Calculate number ways to choose jurors

To determine the number of ways to choose the jurors, we need to find the combinations of selecting 12 jurors out of 30. This is represented as C(30, 12). The formula for combinations is given by: C(n, r) = \(\frac{n!}{(n-r)!r!}\) where n is the total number of items, and r is the number of items being chosen. Using this formula, we will calculate the number of ways to choose 12 jurors from a group of 30 prospective jurors.

Step 2: Calculate the number of ways to choose alternate jurors

After choosing the 12 jurors, there are now 18 prospective jurors remaining. We need to determine the number of ways to choose 2 alternate jurors out of the remaining 18. This can be represented as C(18, 2). Using the combinations formula, we'll calculate the number of ways to choose 2 alternate jurors.

Step 3: Calculate the total number of ways to choose the panel

To find the total number of ways to choose the panel, we need to multiply the number of ways to choose the 12 jurors by the number of ways to choose the 2 alternate jurors. Total ways = C(30, 12) × C(18, 2) Now, we'll calculate the values of C(30, 12) and C(18, 2) and multiply them to get the total number of ways. C(30, 12) = \(\frac{30!}{(30-12)!12!}\) = \(\frac{30!}{18!12!}\) C(18, 2) = \(\frac{18!}{(18-2)!2!}\) = \(\frac{18!}{16!2!}\) Total ways = C(30, 12) × C(18, 2) = \(\frac{30!}{18!12!} × \frac{18!}{16!2!}\) To simplify the expression, we can cancel out the 18! in both the numerator and denominator: Total ways = \(\frac{30!}{12!} × \frac{1}{16!2!}\) Now, we can solve the expression and find the total number of ways: Total ways = \(\frac{30!}{12!} × \frac{1}{16!2!}\) = 5,153,875 Hence, there are 5,153,875 different ways to choose a panel of 12 jurors and 2 alternate jurors from a group of 30 prospective jurors.

Key Concepts

Permutations and Combinations

Understanding the concepts of permutations and combinations is crucial when dealing with scenarios that involve arranging or selecting various elements. Permutations focus on arrangements where the order of selection is important. For example, when we talk about the different orders in which a set of books can be arranged on a shelf, we are considering permutations. In contrast, combinations deal with selections where the order does not matter, such as choosing members for a committee.

In our juris selection exercise, we used combinations because the order in which the jurors are selected does not influence the makeup of the panel. The key formula at play here is the combination formula:C(n, r) = \(\frac{n!}{(n-r)!r!}\).It calculates the number of ways to choose 'r' elements from a larger set of 'n' elements without regard to the order of selection. Here, '!' represents factorial notation, which we'll review in detail in the next section.

Factorial Notation

Factorial notation plays a fundamental role in combinatorics and is especially pivotal in calculating permutations and combinations. A factorial of a non-negative integer 'n', denoted as 'n!', is the product of all positive integers less than or equal to 'n'. This can be expressed as:n! = n × (n-1) × (n-2) × ... × 2 × 1.For example, 5! equals 5 × 4 × 3 × 2 × 1, which is 120. There's a special case where the factorial of zero, 0!, is defined as 1. In our exercise, you encountered factorials when computing the numbers of ways to choose jurors from a pool. Understanding how to calculate and simplify factorials is essential when working with combinations to ensure accurate results. For instance, the cancellation of 18! from the numerator and denominator was a simplification that arose from understanding factorial division.

Probability and Statistics

Probability and statistics are branches of mathematics dealing with data collection, analysis, interpretation, and presentation. In the context of combinatorics, probability concerns itself with the likelihood of various outcomes occurring within a set of all possible outcomes. This often involves the use of permutations and combinations to determine the total number of possible outcomes and the number of ways certain events can occur.

In statistics, combinatorial methods help to analyze the frequency and distribution patterns among a data set. When approaching problems akin to juror selection, this understanding can be extended to calculate the probability of selecting any specific group of jurors. For instance, if we were to calculate the odds of randomly forming a panel that includes a specific prospective juror, we would use the principles from combinatorics. Overall, knowing the fundamentals of probability and statistics enables us to interpret data more effectively and make informed decisions based on likelihood and trends.