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Chapter 11: Problem 16

You are taking a multiple-choice test that has eight questions. Each of the questions has three answer choices, with one correct answer per question. If you select one of these three choices for each question and leave nothing blank, in how many ways can you answer the questions?

Short Answer

Expert verified
There are 6,561 ways to answer the eight questions when each question has three answer choices.

Step by step solution

01

Understanding the Basics

In combinatorics, a branch of mathematics dealing with arrangements and combinations, there is a rule that can solve this problem. When having 'n' choices for one event, and 'm' choices for another independent event, the total combinations for these events occurring simultaneously is \(n \times m\). Here, we are dealing with multiplication, not addition, because these events do not exclude each other.
02

Applying the Rule

In our current case, there are three choices for each of the eight questions, so we use the rule for 'n' and 'm' previously defined to calculate the total combinations. So, for each question, we have three possibilities of answers. Therefore, the total ways to answer the first question is 3. Following the same logic, the total ways to answer each of the remaining seven questions is also 3. So, for eight questions, the total possible ways of answering all of them would be \(3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^8\).
03

Calculating the Result

The final step is to compute \(3^8\), which equals 6,561. Therefore, there are 6,561 ways to answer the eight questions when each question has three answer choices.

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