Problem 38 - Algebra for College Students

Question

Evaluate expression. \(5 !\)

Step-by-Step Solution

Answer

120

Step 1 – Understand the Notation

The notation used here is 'factorial', represented by the exclamation mark (!). For a positive integer n, the factorial (n!) is the product of all positive integers less than or equal to n.

Step 2 – Write Down the Factorial Expression

For the given exercise, the expression is 5!. This can be written out as: 5! = 5 × 4 × 3 × 2 × 1

Step 3 – Perform the Multiplication

Multiply the numbers together in sequence:5 × 4 = 2020 × 3 = 6060 × 2 = 120120 × 1 = 120

Step 4 – Write the Final Result

After performing the multiplications, the result of 5! is 120.

Key Concepts

factorial

A factorial, represented by an exclamation mark (!), is a mathematical function that multiplies a series of descending positive integers. For any positive integer n, the factorial (denoted as n!) is the product of all positive integers from 1 up to n. For example, 5! means multiplying 5 by all the positive integers less than 5:
5! = 5 × 4 × 3 × 2 × 1
One important fact to remember is that the factorial of 0 is defined as 1. This might seem confusing, but it is a commonly agreed-upon rule in mathematics. It's good to practice calculating factorials of small numbers to understand this concept better.

multiplication

Multiplication is one of the basic arithmetic operations. When dealing with factorials, multiplication plays a crucial role since calculating a factorial involves multiplying several numbers together.
For example, in calculating 5!, you have to multiply:
5 × 4 = 20
20 × 3 = 60
60 × 2 = 120
120 × 1 = 120
Each step involves taking the product from the previous calculation and multiplying it by the next integer. This approach ensures the correct result. Understanding multiplication and being able to perform it accurately and quickly is essential for working with factorials and many other math problems.

positive integers

Positive integers are all whole numbers greater than zero. Examples include 1, 2, 3, and so on. When we talk about factorials, we only deal with positive integers.
For instance, in the factorial expression 5!, the numbers we use are all positive integers: 5, 4, 3, 2, and 1. These numbers are then multiplied together in a specific order to find the factorial.
Working with positive integers in factorial expressions is straightforward because they are simple, whole numbers without any fractions or decimals. This makes the multiplication process easier to manage and understand.