Problem 59 - College Algebra

Question

Solve by the method of your choice. A medical researcher needs 6 people to test the effectiveness of an experimental drug. If 13 people have volunteered for the test, in how many ways can 6 people be selected?

Step-by-Step Solution

Answer

The researcher can select 6 people from 13 volunteers in 1716 different ways.

Step 1: Recognize the Problem Type

Recognize that this is a combinatorics problem. Because the order in which the people are selected does not matter, it is a combination problem.

Step 2: Identify the Parameters

Identify the total number of elements, which is the number of volunteers (13), and the number of elements in each combination, which is the number of people selected (6).

Step 3: Apply the Combination Formula

The formula for combinations is \( C(n, k) = n! / [(n-k)!k!] \), where \( n \) is the total number of elements, \( k \) is the number of elements in each combination, and \( ! \) denotes factorial. In this case, it becomes \( C(13, 6) = 13! / [(13-6)!6!] \)

Step 4: Evaluate the Factorials

Evaluate the factorials in the formula: \( 13! = 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6! \), \( 7! = 7 × 6! \), \( 6! = 6 × 5 × 4 × 3 × 2 × 1 \).

Step 5: Perform the Division

Simplify the formula and calculate the answer: \( C(13, 6) = 13! / 7!6! = 1716 \)

Key Concepts

Factorials

Factorials are a fundamental concept in combinatorics and mathematics in general. They are denoted by an exclamation mark (!). A factorial of a number is the product of all positive integers less than or equal to that number. For example, if you have a number 5, the factorial is calculated as:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

Factorials help quantify permutations, which are arrangements of a set where order matters. In many selection problems, factorials are used to determine the number of ways to arrange or select items. This concept is particularly crucial because it simplifies calculations involving large numbers.

A good approach to understanding factorials is to start small and gradually work with larger numbers. You'll often see factorials when calculating probabilities, combinations, or permutations. Once you become comfortable with smaller factorials, higher values will also become easier to manage in calculations.

Combination Formula

The combination formula is central to finding solutions in problems where the order of selection doesn't matter. It allows you to calculate the total number of ways to choose a specific number of items from a larger group without considering the order.

The general formula for combinations is:

• \[ C(n, k) = \frac{n!}{(n-k)!k!} \]

Here:

• \( n \) is the total number of items in the set.
• \( k \) is the number of items to choose.
• \( ! \) indicates factorial as explained earlier.

In our original exercise, we used this formula:

• \[ C(13, 6) = \frac{13!}{7! \times 6!} \]

This calculation provides the number of ways 6 people could be selected from 13 volunteers.

The combination formula simplifies calculations involving large numbers and prevents mistakes often made when manually listing all possible combinations. By using the formula, you quickly arrive at an accurate outcome.

Selection Problems

Selection problems are common in both real-life and theoretical applications. They involve selecting a subset of items from a larger set. The challenge here is usually quantifying the number of possible selections given certain constraints.

For these problems, it's important to consider whether the order of selection matters. If it doesn’t, as in our problem, combinations are the correct approach. If order matters, permutations are used instead.

To approach a selection problem:

• First, decide whether you’re dealing with combinations or permutations based on whether order matters.
• Then, utilize the appropriate formula. For combinations, use the combination formula. For permutations, a different formula applies.
• Identify your total set size (\( n \)) and the size of your selection set (\( k \)).

The considerations and approach will directly affect the complexity and outcome of your solution. Understanding the problem type guides you through the solution process and ensures that the right method is applied to arrive at a numerical answer.