Problem 59 - Algebra and Trigonometry

Question

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

Using the combinations formula, the researcher can select 6 people for the experiment in 1,716 ways.

Step 1: Understand the Combinations Concept

This problem involves combinations, which are about choosing items from a group without taking the order into account. This means that item A followed by item B is the same as item B followed by item A.

Step 2: Apply the Combinations Formula

The combinations formula is denoted as \( C(n, r) = \frac{n!}{r!(n-r)!} \) where \( n \) is the total number of items, and \( r \) is the number of items to choose. Here, \( n = 13 \) (the total number of volunteers), while \( r = 6 \) (the number of people needed for the experiment). So, we can use the formula to find out in how many ways the researcher can select 6 people from 13 volunteers.

Step 3: Calculate the Combination

By substituting \( n = 13 \) and \( r = 6 \) in the combination formula, we get \( C(13, 6) = \frac{13!}{6!(13-6)!} \). '!' denotes factorial which means multiplying all whole numbers from the chosen number down to 1.