Problem 34 - Algebra and Trigonometry

Question

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?

Step-by-Step Solution

Answer

The total number of ways to answer the eight questions is 6,561.

Step 1: Understanding of the Problem and the Rule of Product

In a multiple-choice test, with each question having one correct answer out of three possible options, the number of ways that each question can be answered is 3. If there are 8 such questions, and these choices are independent of each other, then we can use the Rule of Product in probability which states that if there are n1 ways to perform the first operation, n2 ways to perform the second operation, ..., nk ways to perform the kth operation, the operations can be performed in any order, and the number of ways to perform each operation does not depend on the ways the other operations are performed, then there are n1*n2*...*nk ways to perform the operations.

Step 2: Applying the Rule

By the Rule of Product, the total number of ways that the test can be answered is 3*3*3*3*3*3*3*3, or \(3^8\).

Step 3: Calculating the Result

By calculating \(3^8\), we get the total number of ways the questions can be answered on the test is 6,561.