Problem 13 - Algebra for College Students

Question

Yesha has 12 schools to visit this week. In how many different ways can she pick a first, second, and third school to visit on Monday?

Step-by-Step Solution

Answer

1320

Step 1: Determine the number of choices for the first school

Yesha has 12 schools to choose from for her first visit.

Step 2: Determine the number of choices for the second school

After choosing the first school, there are 11 schools remaining for the second choice.

Step 3: Determine the number of choices for the third school

After choosing the first and second schools, there are 10 schools remaining for the third choice.

Step 4: Calculate the total number of ways

Multiply the number of choices for each step: \[ 12 \times 11 \times 10 = 1320 \]

Key Concepts

Factorial

The concept of factorial is key in understanding permutations. Factorial, denoted by an exclamation mark (!), is a product of all positive integers up to a given number. For instance, the factorial of 4 (written as 4!) is calculated as follows:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]
Factorials are essential because they help us compute the total number of ways to arrange objects. Though we didn't use a full factorial in the exercise, understanding it is vital for tackling more complex problems in combinatorics.

Combinatorics

Combinatorics is the study of counting and arrangement. It encompasses a wide range of concepts including permutations and combinations. In this exercise, we focus on permutations, where the order of selection matters. Combinatorics helps us understand how to count different ways to arrange a set of items. By learning combinatorics, we can solve various real-world problems associated with arranging, selecting, and organizing items.

Sequential Choices

Sequential choices involve making decisions one after another. Each choice affects the next. In our exercise, Yesha makes sequential choices by selecting three schools out of twelve.

• First, she picks the first school with 12 options.
• Next, she picks another school from the remaining 11 schools.
• Finally, she picks the third school from the 10 schools left.

These steps illustrate the importance of recognizing how each choice reduces the number of remaining options. Sequential choices are a common scenario in probability and combinatorics.

Multiplicative Principle

The multiplicative principle states that if there are multiple steps in a process, the total number of outcomes is the product of the number of choices at each step. In Yesha's problem:

\[ Total = 12 \times 11 \times 10 \]

We multiply the number of choices for each step, resulting in \[ 12 \times 11 \times 10 = 1320 \]. This principle helps simplify many complex problems in combinatorics by breaking them down into manageable steps and multiplying the number of options at each stage. It's a fundamental concept that, once mastered, can help solve a wide variety of counting problems.