Problem 11 - Algebra for College Students

Question

Randall has homework in mathematics, history, art, literature, and chemistry but cannot decide in which order to attack these subjects. How many different orders are possible?

Step-by-Step Solution

Answer

Randall can arrange his homework in 120 different ways.

Step 1: Understand the Problem

Randall has 5 different subjects to complete: mathematics, history, art, literature, and chemistry. You need to find the number of different ways to arrange these 5 subjects.

Step 2: Identify the Total Number of Items

In this problem, the total number of subjects is 5. We need to determine the number of different ways to arrange or order these 5 subjects.

Step 3: Use the Permutation Formula

Permutations are used when the order of items matters. The formula to find the number of permutations of n distinct items is given by \[ n! = n \times (n-1) \times (n-2) \times \text{...} \times 2 \times 1 \] Here, \( n \) is 5 because there are 5 subjects.

Step 4: Calculate the Permutation

Apply the values into the permutation formula: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 \] Calculate the product: \[ 5! = 120 \]

Step 5: Conclusion

There are 120 different ways Randall can arrange his homework for the 5 subjects.

Key Concepts

factorial

A factorial is a concept in mathematics symbolized by an exclamation mark (!). The factorial of a number, often denoted as n!, is the product of all positive integers less than or equal to n. For example, the factorial of 5 is calculated as:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
Factorials grow very fast. Even for small numbers, the factorial values can be quite large. Factorials are essential in various fields of mathematics, including combinatorics, algebra, and calculus. They help count different arrangements or permutations of a set of items.
If you imagine arranging 5 different books on a shelf, the number of unique ways to do that is given by 5!. This same idea is applied in our exercise with Randall's homework.

permutation formula

Permutations refer to arrangements where the order of items matters. The permutation formula helps determine how many different ways a set of n items can be ordered. It is represented by the factorial of the number of items: \[ P(n) = n! = n \times (n-1) \times (n-2) \times \text{...} \times 2 \times 1 \]
If you have 5 subjects like in Randall's case, the permutation formula tells us how many unique sequences these subjects can be arranged in: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
This formula is very powerful for problems involving ordering. It applies not just in homework sequences, but in various scenarios like seating arrangements, password possibilities, and more.

combinatorics

Combinatorics is a branch of mathematics dealing with combinations and permutations of sets of elements. It helps solve problems related to counting, arrangement, and selection. It plays a critical role in fields like computer science, cryptography, and operations research.
In our exercise, combinatorics helps us figure out the number of ways Randall can order his 5 subjects. By understanding permutations and applying the factorial concept, we use combinatorics to find our answer.
Some key areas within combinatorics include:

• Counting techniques
• Graph theory
• Combinatorial design
• Probability

For students, mastering the basics of combinatorics can make many counting problems easier to handle, whether in homework or real-life scenarios.