Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

A true-or-false test has 20 questions. (a) In how many different ways can the test be completed? (b) In how many different ways can a student answer 10 questions correctly?

Step-by-Step Solution

Answer

(a) 1,048,576 ways to complete the test. (b) 184,756 ways to answer 10 questions correctly.

Step 1: Understand Total Possibilities

For part (a), we need to consider how many different ways a student can answer a true-or-false question. Each of the 20 questions has 2 possible options: true or false. Using the principle of multiplication, the total number of different ways to complete all questions is given by multiplying the number of choices (2) for each question across all 20 questions. This can be expressed as \(2^{20}\).

Step 2: Calculate Total Ways to Complete the Test

Compute \(2^{20}\) to find the total number of different ways to answer the 20 questions.Compute: \[2^{20} = 1,048,576\]Thus, there are 1,048,576 different ways to complete the test.

Step 3: Understand Correct Answers Possibility

For part (b), we consider how many ways a student can answer 10 questions correctly out of 20. This is a combination problem where we choose 10 questions to answer correctly out of 20 questions in total. The number of combinations is represented by the binomial coefficient \(\binom{n}{r}\), which stands for choosing \(r\) objects from \(n\) objects without regard for the order.

Step 4: Calculate Combinations for Correct Answers

Use the formula for combinations \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\) to find how many ways 10 questions can be answered correctly.Apply the values:\[\binom{20}{10} = \frac{20!}{10! \times 10!}\]Calculate \(\binom{20}{10}\) using a calculator or simplified combinatorial calculation.The value is 184,756. Therefore, there are 184,756 different ways to answer 10 questions correctly.

Key Concepts

True-or-False Test

A true-or-false test is a common type of assessment where each question offers two possible answers: "true" or "false." This simplicity allows for quick decision-making but also introduces a statistical approach to understanding the number of possible answer combinations.
Each question's binary nature means that for any given question, a student has two choices. Therefore, for a series of questions, the total number of possible answers is calculated using the principle of multiplication by multiplying the number of choices per question.
In the context of a 20-question true-or-false test, there are two choices per question, leading to a possible total number of combinations equal to raising the number of choices (2) to the power of the number of questions (20), or 2^20. This results in a total of 1,048,576 different ways to complete the entire test.

Binomial Coefficient

The binomial coefficient is a fundamental concept in combinatorics, representing the number of ways to choose a subset of items from a larger set, without regard to the order of selection.
It's important in situations like determining the number of ways to get a certain number of correct answers on a true-or-false test.
For instance, if a student wants to answer 10 questions correctly out of 20, the calculation of combinations can be represented as \( \binom{20}{10} \).
This coefficient is calculated using the formula:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

where \( n \) is the total number of items, and \( r \) is the number of items to choose.
Here, \( 20! \) is the factorial for the total questions, and the factorial solves the problem of possible arrangements. The calculation of \( \binom{20}{10} \) gives a result of 184,756, meaning there are 184,756 different ways for the student to get exactly 10 questions right.

Principle of Multiplication

The principle of multiplication, often called the rule of product, is a key concept in combinatorics used to calculate the number of possible outcomes when there are multiple independent choices.
It states that if one event can occur in \( m \) ways and a second, independent event can occur in \( n \) ways, then there are \( m \times n \) total ways for both events to occur.
This principle is applied extensively in scenarios like our true-or-false test, where each question (event) can be answered in two ways independently of other questions.
By multiplying the number of choices for each event, you determine the total number of combinations possible. For a 20-question test, applying the principle of multiplication results in a total of \( 2^{20} \), or 1,048,576, illustrating how independent options rapidly increase the complexity and variety of combinations.