Problem 7 - Algebra 2

Question

For a college application, Macawi must select one of five topics on which to write a short essay. She must also select a different topic from the list for a longer essay. How many ways can she choose the topics for the two essays?

Step-by-Step Solution

Answer

There are 20 ways Macawi can choose the topics for the two essays.

Step 1: Understand the Problem

Macawi has five different topics to choose from for her essays: one for a short essay and one for a longer essay. Importantly, the topics must be different for each essay.

Step 2: Choose a Topic for the Short Essay

There are 5 possible topics that Macawi can choose for her short essay. So, she has 5 options for this choice.

Step 3: Choose a Different Topic for the Long Essay

After selecting a topic for the short essay, only 4 topics remain for the longer essay since the topics must be different. Therefore, there are 4 options for the long essay.

Step 4: Calculate the Total Number of Combinations

To find the total number of ways Macawi can choose topics for the two essays, multiply the number of options for the short essay (5) by the number of options for the long essay (4): \[ 5 \times 4 = 20 \]

Key Concepts

Permutations

Permutations are concerned with the arrangement of objects in a specific order. In the exercise, Macawi has to pick topics for her essays such that she writes a different essay for each topic. This situation is a classic permutation problem because the order in which she selects the topics matters due to the requirement for different topics. The formula for permutations of selecting and arranging \( r \) items from \( n \) distinct items is given by the expression \( n \times (n-1) \times (n-2) \times...\) up to \( r \) terms. Here, since Macawi has to select one topic for the short essay and another for the long essay, it is a permutation of selecting 2 topics from 5 available topics, calculating as \( 5 \times 4 \). Thus, there are 20 possible ways she can arrange the essays, considering the order of chosen topics.

Combinations

Combinations involve selecting objects where the order does not matter. However, this core concept is still valuable to understand within the context, even if the problem specifically dealt with permutations. It helps differentiate when combinations are useful, like when choosing a committee or picking lottery numbers, where it doesn't matter which comes first. The formula for combinations is \( \frac{n!}{r!(n-r)!} \), which signifies how to compute the number of ways to choose \( r \) objects from \( n \) without concern for the order. Though not necessary for solving the given problem, understanding combinations is key to distinguishing contexts where each method is applicable. Recognizing that this exercise discerningly calls for permutations can deepen comprehension of both concepts.

Problem-solving strategies

Solving problems in combinatorics often involves breaking down the problem into smaller parts. For Macawi's exercise, it might initially seem complex, but by dissecting the task into choosing topics sequentially, it becomes manageable.
This is a fundamental strategy:

• **Understand the requirements:** Make sure the problem's constraints are clear, like choosing different topics here.
• **Step-by-step selection:** Break the decision process into steps, as Macawi does by selecting first for the short and then the long essay.
• **Calculate systematically:** Count available options at each stage, multiplying choices to accumulate total outcomes.

These strategies alleviate overwhelming problems by structuring a path to reach the solution efficiently. Whether dealing with combinations, permutations, or other types of counting problems, methodical approaches make problem-solving less daunting and more intuitive.