Factorials - Online Tutor, Practice Problems & Exam Prep

Factorials, denoted by an exclamation point, represent the product of all whole numbers from a given number down to 1. For example, 4! equals 4⁢3⁢2⁢1, which equals 24. The relationship n! = n⁢(n-1)! simplifies calculations. Notably, 0! equals 1, establishing a foundational understanding of factorials in combinatorics and probability.

Factorials

Hey, everyone. As you work through different problems, you may come across some that have a number or a variable or an expression followed by an exclamation point. And it's not just that we're really excited to write that number because this exclamation point actually represents a specific type of operation called a factorial. Now, this might seem a bit strange at first because you're not used to using punctuation as a math operation. But here I'm going to walk you through exactly what a factorial is and how to simplify different factorial expressions. And soon, this will be just like any other operation, like addition or subtraction. So, let's go ahead and get started.

Now, I said that a factorial was a specific type of operation, and the operation that it represents is multiplying all whole numbers from some specific number down to 1. So, for example, if I took 4 factorial, I would start at 4 and then count down to 1, multiplying all of those whole numbers along the way. So, counting down to 1, I get 4, 3, 2, 1, and multiplying all of those numbers together would give me the factorial of 4. A factorial is always going to be represented with an exclamation point, whether it be in sequences, series, combinatorics, or probability.

So, now that we know what a factorial is, let's go ahead and calculate some. Now, I'm gonna skip 0 factorial for now because it's a little bit strange, but we're gonna come back to it. So, let's go ahead and start with 1 factorial. Now, for 1 factorial, I can't really count down anywhere, so 1 factorial is simply going to be equal to 1. Now looking at 2 factorial, I can count down here. So, I'm gonna go 2 times 1, and 2 factorial is then going to be equal to that, which is 2. Now for 3 factorial, I'm going to count down from 3 multiplying 3 times 2 times 1, which will give me a value of 6. So, 3 factorial is equal to 6.

Now looking here at 4 factorial, I'm going to start at 4, which we already saw this, and multiply times 3 times 2 times 1. Now multiplying all of this out will give me a value of 24. Then looking at 5 factorial, now I'm left to multiply 5 times 4 times 3 times 2 times 1 to give me a value of 120. Now, something that you may have noticed here is all we're really doing is taking some new number and then multiplying it by the previous factorial. So, for 6 factorial, I would simply take my new number of 6 and multiply it by the previous factorial, which is 5, so 5!. Now, 6 times 5! is going to give me a value of 720 because all we did was multiply this previous factorial, 120, by 6.

Now to generalize this a little bit more, if I have any number n, I'm simply going to take that number and multiply it by (n-1)!, just like we saw for 6 × 5!. This is going to allow us to simplify some factorial expressions a bit easier, so let's go ahead and work through some examples and see exactly how that's going to work for us.

Now looking at our first example here, we have 4 times 3!. But using this formula that we just learned, this is really just the same thing as 4 factorial, which we already know the value of. Up here we saw that 4! was equal to 24, so that's going to be our final answer here, that 4 times 3! is equal to 24.

So let's look at our second example here. We have 100!/99!. Now your first instinct here might be to go ahead and start multiplying all of these out, counting down to 1, 100 times 99 times 98 times 97, and so on. But we don't have to do that because we have this formula. So, I can actually just rewrite 100! using this formula, and let's see what happens. 100! is really just 100 times 99!, and if I'm dividing all of that by 99!, well, this is just going to cancel out. So, that 99! on the top and the bottom just completely goes away, leaving me with a simple answer of 100. No crazy multiplication needed.

Let's look at our final example here. Here we have 1 minus 1 factorial. Now, you might be thinking that isn't this just 0 factorial? And you're right. It is. But remember, I said 0 factorial was a bit weird. So, let's think about this a little bit deeper using this formula here. This 1 minus 1 factorial looks kind of similar to this (n-1)! function. So, if n was equal to 1 and we plugged that into this equation, we're going to see that 1 factorial is equal to 1 times (1-1)!. Now here we see this 1 minus 1 factorial, which is exactly what we're looking for here. So, let's go ahead and simplify this a bit more. Well, I know that one factorial is just equal to 1, so this is really just 1 and that's equal to I have 1 times 1 minus 1 factorial. Now, one times anything is just that thing. So, this leaves me that 1 is equal to 1 minus 1 factorial. And we already have our answer here, that 1 minus 1 factorial is equal to 1, which tells us that 0 factorial is actually just equal to 1. And we can fill in that final value in our table up here that 0 factorial is 1. So now that we know what a factorial is and how to simplify different expressions, let's get some more practice. Thanks for watching, and I'll see you in the next one.

Evaluate the expression.

$\frac{12}{4!}$

• A. $\frac14$
• B. $\frac12$
• C. $2$
• D. $3$

Evaluate the expression.

$\frac{9!}{7!}$

• A. 2!
• B. 63
• C. 72
• D. 98

Evaluate the expression.

$\frac{16!}{12!\cdot4!}$

• A. 0
• B. 1
• C. 1,820
• D. 43,680

Write the first 4 terms of the sequence $a_{n}=n^2\cdot\left(n-1\right)!$

• A. {1, 4, 18, 96}
• B. {1, 2, 6, 24}
• C. {0, 4, 18, 96}
• D. {0, 2, 6, 24}

Here’s what students ask on this topic:

What is a factorial and how is it calculated?

A factorial, denoted by an exclamation point (!), represents the product of all whole numbers from a given number down to 1. For example, 4! (read as '4 factorial') is calculated as 4 × 3 × 2 × 1, which equals 24. The general formula for a factorial is n! = n × (n-1)!, where n is a positive integer. This means that each factorial is the product of the number and the factorial of the number one less than it. Notably, 0! is defined to be 1.

Why is 0 factorial equal to 1?

0 factorial (0!) is defined to be 1. This might seem counterintuitive at first, but it makes sense when considering the definition of factorials and their use in combinatorics. The factorial of a number n is the product of all positive integers up to n. For 0!, there are no positive integers to multiply, so by convention and for mathematical consistency, it is defined as 1. This definition helps in simplifying expressions and maintaining the consistency of formulas, such as the binomial coefficient.

How do you simplify factorial expressions?

To simplify factorial expressions, you can use the relationship n! = n × (n-1)!. For example, to simplify 100! / 99!, you can rewrite 100! as 100 × 99!, which allows the 99! terms to cancel out, leaving you with 100. This method avoids the need for extensive multiplication and makes calculations more manageable. Another example is 4 × 3!, which can be simplified to 4! because 4! = 4 × 3 × 2 × 1.

What are some common applications of factorials?

Factorials are commonly used in combinatorics, probability, and series. In combinatorics, they help calculate permutations and combinations, which are ways to arrange or select items. For example, the number of ways to arrange n items is given by n!. In probability, factorials are used to determine the likelihood of different outcomes. They also appear in the expansion of series, such as the Taylor series, where factorials help in calculating the coefficients of the terms.

How do you calculate large factorials efficiently?

Calculating large factorials directly can be computationally intensive. To do this efficiently, you can use properties of factorials, such as the relationship n! = n × (n-1)!, to break down the calculation into smaller parts. Additionally, using software tools or programming languages that support large integer calculations can help. For example, Python has a built-in factorial function in its math library that can handle large numbers efficiently. Another approach is to use logarithms to transform the multiplication into addition, which is computationally simpler.