Problem 31 - Algebra for College Students

Question

How many ways are there to mark the answers to a test that consists of 10 true-false questions followed by 10 multiple-choice questions with 5 options each?

Step-by-Step Solution

Answer

There are 10,000,000,000 ways to mark the answers.

Step 1 - Determine the Choices for True-False Questions

Each of the 10 true-false questions has 2 possible answers (True or False). To find the total number of ways to answer these questions, use the formula for the number of combinations: \(2^{10}\)

Step 2 - Calculate Total Ways for True-False Questions

Calculate \(2^{10}\). This gives the number of ways to answer the 10 true-false questions: \(2^{10} = 1024\)

Step 3 - Determine the Choices for Multiple-Choice Questions

Each of the 10 multiple-choice questions has 5 possible options (A, B, C, D, E). To find the total number of ways to answer these questions, use the formula for the number of combinations: \(5^{10}\)

Step 4 - Calculate Total Ways for Multiple-Choice Questions

Calculate \(5^{10}\). This is the number of ways to answer the 10 multiple-choice questions: \(5^{10} = 9,765,625\)

Step 5 - Calculate the Total Number of Ways to Answer the Test

Multiply the number of ways to answer the true-false questions by the number of ways to answer the multiple-choice questions: \( 1024 \times 9,765,625 = 10,000,000,000\)

Key Concepts

True-False Questions

True-false questions are a common type of question format in exams. Each question offers two possible answers: True or False. Understanding the number of possible combinations can be quite useful.
For instance, if you have 10 true-false questions, each one can be answered in 2 ways. To find out the total number of ways to answer all these questions, you use exponential notation:
\(2^{10}\)
Here, the base (2) represents the two possible answers, and the exponent (10) shows the number of questions. So, the total number of ways to answer is:
\[2^{10} = 1024 \]
Thus, for 10 true-false questions, there are 1,024 different combinations of answers.

Multiple-Choice Questions

Multiple-choice questions are another frequent format in tests. Each question has several answer options. In this specific problem, there are 5 options (A, B, C, D, E) for each multiple-choice question.
To determine the number of combinations for these types of questions, you again use exponential notation. For 10 multiple-choice questions, where each has 5 options, you calculate the total number of ways like this:
\(5^{10}\)
Here, the base (5) is the number of choices for each question, and the exponent (10) is the number of multiple-choice questions.
So, \[5^{10} = 9,765,625 \]
This means there are 9,765,625 different ways to answer the 10 multiple-choice questions.

Number of Combinations

Combining these types of questions allows you to calculate the overall number of ways to answer the test. If your test includes both true-false and multiple-choice questions, you multiply the number of ways to answer each type.
From our example:
\(2^{10} = 1024\) ways to answer the true-false questions
\(5^{10} = 9,765,625\) ways to answer the multiple-choice questions
To find the total number of combinations for the entire test, you multiply these two results:
\[1024 \times 9,765,625 = 10,000,000,000 \]
Therefore, there are 10 billion different ways to complete a test with 10 true-false and 10 multiple-choice questions.
Understanding these calculations can help in future problem-solving and provides a foundational skill in combinatorial analysis.