Problem 51 - College Algebra

Question

In Exercises \(49-58,\) 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

There are 1,716 ways for the medical researcher to select 6 people from a group of 13 volunteers.

Step 1: Understand the Problem

This problem is about selecting 6 people from a group of 13 volunteers. The order in which these people are selected does not matter, so this is a combination problem. Therefore, the formula for calculating combinations will be used.

Step 2: Apply the Combination Formula

The combination formula is \[ C(n, k) = \frac{n!}{k!(n-k)!} \], where \( n \) is the total number of options, \( k \) is the number of options chosen, and \( ! \) denotes factorial, which is the product of all positive integers less than or equal to that number. In this case, \( n = 13 \) and \( k = 6 \). Plug these values into the combination formula: \[ C(13, 6) = \frac{13!}{6!(13-6)!} \].

Step 3: Calculate Factorials

First, calculate the factorial of 13, then the factorial of 6, and then the factorial of 7 (13-6). Use these values to find the value of \[ \frac{13!}{6!(13-6)!} \]

Step 4: Divide

Divide \( 13! \) by \( 6! \times 7! \) to find the number of combinations.

Key Concepts

Factorial

In mathematics, a factorial is a fundamental concept used to calculate the number of ways items can be arranged or combined. The factorial of a number, represented as

• \( n! \), is the product of all positive integers less than or equal to \( n \).
• For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
• By definition, \( 0! \) is equal to 1.

Factorials are crucial in calculating combinations and permutations because they give us the total number of possible ways to arrange a set of items.In the given exercise, factorials are used to determine the number of ways to choose 6 people from 13 volunteers based on the combination formula. By calculating the factorials of 13, 6, and 7, you can find the number of selections possible using the formula for combinations.

Permutations

Permutations refer to the arrangement of items in a specific order. This concept differs from combinations, where the order of selection does not matter.

• A permutation is used when the arrangement or sequence of items is important.
• The formula for permutations of \( n \) items taken \( k \) at a time is represented by \( P(n, k) = \frac{n!}{(n-k)!} \).

It's important to distinguish between permutations and combinations within mathematical problems, as confusing them could lead to incorrect solutions. While the original exercise focused on combinations, understanding permutations helps deepen your comprehension of different types of groupings and arrangements possible in mathematical problem-solving.

Mathematical Problem-Solving

Solving mathematical problems involves understanding key concepts and applying appropriate problem-solving techniques. In the exercise provided, several steps were used to solve the problem of selecting 6 people from 13 volunteers:

• Understanding the Problem: This involves identifying the type of problem (combinations in this case) and the formula needed.
• Applying the Combination Formula: Using the formula \( C(n, k) = \frac{n!}{k!(n-k)!} \), the problem is articulated in mathematical terms.
• Calculating Factorials: Specific calculations were needed to determine the values of \( 13! \), \( 6! \), and \( 7! \), which are crucial for solving the combination formula.
• Performing the Division: By dividing \( 13! \) by \( 6! \times 7! \), the number of ways to select 6 people was finally calculated.

Each step aids in simplifying the process, ensuring you derive the correct solution. Problem-solving in mathematics often involves breaking tasks into smaller, more manageable parts while systematically applying mathematical principles.