In this explainer, we will learn how to find the factorial of any number , which is the product of all integers less than or equal to and greater than or equal to one, and we will learn how to find factorials to solve problems.
When considering the number of different 4-digit numbers that we can form from the digits 3, 5, 7, and 9, we find that there are a total of distinct possible numbers. More generally, if we want to know how many ways we can rearrange a set of items, we find that the total number is . This calculation, of finding the product of all of the positive integers less than or equal to , comes up sufficiently regularly in various areas of mathematics that mathematicians have a given it a name: the factorial of .
Definition: Factorial
The factorial of a positive integer is the product of all the positive integers less than or equal to . We use the notation , which is read as factorial, to denote the factorial. Hence,
We define the factorial of zero to be equal to one; that is, .
From the definition, it is not difficult to see that, for any integer ,
In many ways, this is the key property of the factorial and, as we will see, it will be applied time and again to solve problems involving factorials.
Let us practice computing a factorial in our first example.
Example 1: Finding Factorials
Evaluate
Answer
Recall that the definition of is the product of all the positive integers less than or equal to , or equivalently
Hence,
There are two factorials which equal 1, which are and . We will use this property of factorials to solve the next example.
Example 2: Solving Problems Involving Factorials of Zero
Find the solution set of .
Answer
We might be tempted to reason as follows: Since , . Hence, . Unfortunately, this is not the complete answer. Instead, we need to remember that and zero is not the only number whose factorial is equal to one. In particular, the factorial of one is also equal to one: . Hence, to solve , we have to consider both cases: where and where . Hence, we find that and are both possible solutions. Therefore, the solution set is {26, 27}.
Most scientific calculators have a button for calculating the factorial of a number. In examples like the one above, it would be completely legitimate to simply use a calculator to evaluate the expressions. However, this is not always possible. In fact, factorials become large so quickly that most calculators are unable to evaluate factorials of numbers larger than 69. However, this does not mean that we are unable to work with them. Instead, using the properties of factorials will enable us to solve problems which involve numbers that are too large for our calculators.
In our next example, we will use properties of factorials to evaluate the given expression involving factorials.
Example 3: Properties of Factorials
Without using a calculator, evaluate the expression .
Answer
Using the following property of the factorial: we see that
Hence,
We can also apply this property when working with expressions involving the factorial of an unknown number.
Let us consider an example along this line.
Example 4: Simplifying Expressions Involving Factorials
Simplify .
Answer
Using the following property of factorials:
we can rewrite
Applying the same property again, we can write
Substituting this in the given expression we have
Canceling the common terms in the numerator and denominator, we get
Let us consider an example where we use properties of factorials to solve an equation involving factorials.
Example 5: Using the Properties of Factorials to Solve Problems
Find the value of which satisfies the equation .
Answer
Using the following property of factorials: we can rewrite
Substituting this into the given equation, we have
Canceling the common factors in the numerator and denominator, we can rewrite this as
Rearranging gives .
Let us consider another example where we use properties of factorials to solve an equation involving factorials.
Example 6: Solving Factorial Equations
Find the solution set of .
Answer
We begin by multiplying both sides of the equation by which gives
Using the following property of factorials: we can rewrite and Substituting these into the equation in the numerators, we have
Canceling the common factors in the numerators and denominators, we can rewrite this as
Expanding the brackets, we have
Rearranging, we arrive at the quadratic
By factoring, or applying the quadratic formula, we can express this as
Hence, or . Since factorials are only defined for nonnegative integers, we can discard the solution . Hence, the solution set of the equation is {7}.
Thus far, we have been able to use the properties of factorials to simplify equations and isolate any unknowns as solutions to linear or quadratic equations. However, these techniques will not help us when we need to find an unknown number given its factorial. For this, we use the fact that a factorial is the product of positive integers less than or equal to a particular number. Therefore, given a number, we can successively divide by consecutive positive integers until we are left with an integer. The next example will demonstrate this process.
Example 7: Finding an Unknown Number Given its Factorial
Find the value of such that .
Answer
Since a factorial is a product of consecutive positive integers, we can divide 720 by consecutive positive integers as follows. Starting with 1: since we can rewrite
Next we divide 720 by 2 which gives us
Hence,
Dividing 360 by 3 gives 120. Therefore, we can write
Similarly, by dividing 120 by 4, we get 30. Hence,
Finally, dividing 30 by 5, we get 6 which gives
Using this process, we have expressed 720 as a product of the first 6 consecutive integers. Therefore, we have . We can write this more succinctly as
Hence, our final answer is .
We finish by looking at another example where we can apply all the techniques we have learned to solve one final factorial problem.
Example 8: Problem-Solving with Factorials
Find the value of such that .
Answer
First let us consider the value 5βββ040. Since we have the product of a factorial and an integer on the left-hand side of the equation, we would like to express 5βββ040 as a factorial or as the product of a factorial and another integer. To do this, we can divide it consecutively by the natural numbers as follows:
Now we can consider the other side of the equation. Remember that for a positive integer
Currently, we do not have two consecutive numbers and to apply this formula. However, by multiplying and dividing by 8, we can produce two consecutive numbers and and apply the formula as follows:
Therefore,
Hence, .
Let us finish by recapping a few important concepts.
Key Points
- The factorial of a positive integer is defined as the product of all positive integers less than or equal to . We write it as .
- The key property of the factorial is that . Using this, we can often simplify expressions involving factorials and solve factorial equations.
- When trying to find an unknown integer given its factorial, we divide by consecutive positive integers.
