Problem 25 - Algebra 2

Question

School Mr. Hopkins is giving a five-question true-false quiz. How many ways could a student answer the questions with three trues and two falses?

Step-by-Step Solution

Answer

There are 10 ways to answer with three trues and two falses.

Step 1: Understand the Problem

We need to find the number of ways a student can answer a five-question true-false quiz with exactly three questions answered as 'true' and two as 'false'.

Step 2: Identify Combinatorial Scenario

Since only the number of true (T) and false (F) answers matter, and we're finding ways to achieve a certain arrangement, this is a combination problem. We want to choose 3 questions to answer 'true' out of the 5 available questions.

Step 3: Apply Combination Formula

Use the combination formula to determine how many ways to choose 3 questions to be true: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]Here, \(n = 5\) and \(r = 3\).

Step 4: Calculate the Combination

Calculate the number of combinations: \[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4 \times 3!}{3! \times 2!} = \frac{20}{2} = 10 \]

Step 5: Conclude Result

There are 10 different ways for a student to answer the questions with three trues and two falses.

Key Concepts

Combination Formula

Combinatorics often involves finding out how combinations can be made from a larger set of items. The combination formula is a mathematical way to determine possible arrangements or selections where order does not matter. This formula is represented as:\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]where:

• \(n\) is the total number of items available.
• \(r\) is the number of items you want to choose.
• \(!\) denotes factorial, a product of all positive integers up to that number.

Let's apply this to our problem, where we select positions for 3 true answers out of 5 available questions. By using this formula, we are choosing all possible sets of 3 positions from a set of 5 positions, without considering the sequence those positions are used.

True-False Quiz

In a true-false quiz, each question has only two possible responses: true or false. This type of quiz is straightforward because, for any given question, there is no in-between option.
The problem at hand is understanding how many different responses a student can provide while maintaining specific conditions - in this case, exactly three answers as true and two as false. Exploring this scenario involves combinatorial calculations to ensure all conditions are met.
A true-false scenario is a classic illustration of binary choices in combinatorics, which often serve as the foundation for understanding more complex probability and statistics concepts.

Factorial

Factorials are critical in calculating combinations. The notation \(n!\) (called "n factorial") represents the product of all integers from 1 to \(n\). For instance, \(5!\) equals \(5 \times 4 \times 3 \times 2 \times 1\).
Factorials grow very quickly with larger numbers, and they are used in the denominator of the combination formula to adjust for overcounting arrangements. This adjustment allows us to calculate only the unique combinations instead of all possible permutations.

• This is particularly useful in scenarios where order does not matter, as in our true-false quiz example.
• Understanding how to manipulate factorials is a fundamental skill in most combinatorics and probability problems.

Problem Solving Steps

Solving combinatorial problems effectively requires a systematic approach. Here's how you can systematically tackle similar problems:

• Understand the Problem: Clearly define what is being asked. Here, determining how many arrangements of true and false answers meet the criteria.
• Identify the Combinatorial Nature: Decide whether the situation involves permutations or combinations. In this case, it's a combination because order doesn't matter.
• Apply the Formula: Insert values into the combination formula and simplify using factorials.
• Calculate and Verify: Perform the arithmetic, double-check the steps, and ensure the logic aligns with the problem statement.

Following these structured problem-solving steps helps to avoid mistakes and ensure a clear path to finding solution.