Zero factorial is defined as 1 by convention. While it might seem strange, this definition makes permutation and combination formulas work correctly. For instance, there is exactly one way to arrange zero objects (by doing nothing), which aligns with the definition 0! = 1.

What is the fastest way to calculate a large factorial?

Instead of multiplying all the numbers, look for opportunities to cancel. For example, \frac{10!}{8!} = 10 \times 9 rather than computing the full factorials. For factorials larger than about 12!, a calculator or computer is the practical choice since the results exceed millions.

Can you use factorial on negative numbers or decimals?

No; in elementary algebra, factorial is only defined for non-negative integers (0, 1, 2, 3, …). Expressions like (-3)! or (2.5)! have no meaning in this context. Advanced mathematics does define factorial for other values, but that is beyond Algebra II.

When do I need to use factorial in real life?

Factorials appear whenever you count arrangements or orderings: seating arrangements for events, scheduling tasks, arranging objects in a display, and many probability and statistics problems. Any time you ask 'how many different orders are possible?', factorial is likely the tool you need.

Factorial

Factorial multiplies all positive integers up to n to find permutations. Use it to count the number of ways to arrange distinct objects in order.

n! = n(n-1)(n-2)\cdots 1, \quad 0! = 1

Solve a problem with Factorial

Type the problem. The solver will use Factorial where Factorial is the right tool, and tell you when it is not.

What each symbol means

What Factorial takes

Symbol	Meaning
$n$	The count of items you are arranging; it must be a non-negative integer (0, 1, 2, 3, …). If you misread $n$ as the items themselves rather than their count, you will calculate the wrong result.

When to use it

You should use factorial when you need to count the number of different ways to arrange a fixed set of distinct items in order.

Level

Usually taught in: Algebra II

Worked examples

1. Calculate 5 factorial

Problem

Find

5!

$5! = 5 \times 4 \times 3 \times 2 \times 1$
Write the definition of factorial: multiply all positive integers from 5 down to 1.
$5 \times 4 = 20$
Multiply the first two numbers.
$20 \times 3 = 60$
Multiply the result by the next integer.
$60 \times 2 = 120$
Continue multiplying by each remaining integer.
$120 \times 1 = 120$
Multiply by 1, which does not change the result.

Answer: $5! = 120$

This is a straightforward application of the factorial definition. We multiply together all positive integers from 5 down to 1 in sequence.

2. Simplify a ratio of factorials

Problem

Evaluate

\frac{7!}{5!}

$7! = 7 \times 6 \times 5!$
Recognize that $7!$ can be rewritten as $7$ times $6$ times $5!$ using the definition of factorial.
$\frac{7!}{5!} = \frac{7 \times 6 \times 5!}{5!}$
Substitute the expanded form into the fraction so the smaller factorial appears in both numerator and denominator.
$7 \times 6 = 42$
The $5!$ cancels from numerator and denominator, leaving only $7 \times 6$ to multiply.

Answer: $\frac{7!}{5!} = 42$

Dividing one factorial by another is a common pattern in permutation problems. By recognizing that $7! = 7 \times 6 \times 5!$ , you can cancel the smaller factorial and avoid computing huge numbers.

3. Arrange concert tickets by price tier

Problem

You just sold 4 premium concert tickets and 3 standard concert tickets. For a promotional display, you need to arrange all 7 tickets in a row, with all premium tickets grouped first and all standard tickets grouped last. How many different orders are possible?

$4! = 4 \times 3 \times 2 \times 1 = 24$
There are 4 premium tickets to arrange within their section, which gives $4!$ possible orders.
$3! = 3 \times 2 \times 1 = 6$
There are 3 standard tickets to arrange within their section, which gives $3!$ possible orders.
$4! \times 3! = 24 \times 6 = 144$
By the multiplication principle, each arrangement of premium tickets can be paired with any arrangement of standard tickets, so we multiply the counts.

Answer: $144 \text{ different orders}$

This problem uses the multiplication principle together with factorials. Since the 4 premium tickets can be arranged in $4!$ ways and the 3 standard tickets in $3!$ ways independently, the total number of arrangements is their product.

Common mistakes

Where Factorial usually goes wrong

The mistake	Why it is wrong	The fix
Computing $5! = 5 + 4 + 3 + 2 + 1 = 15$	Factorial uses multiplication to combine the integers, not addition.	$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
Stopping early, such as writing $5! = 5 \times 4 \times 3 = 60$	The definition of factorial requires multiplying all positive integers from $n$ down to 1, not just some of them.	$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
Writing $0! = 0$	Zero factorial is defined to equal 1 by convention, even though this may seem counterintuitive.	$0! = 1$ (by definition)

Tips and when to use something else

Factorials grow extremely fast; simplify using division before computing: $\frac{10!}{8!} = 10 \times 9 = 90$ is much easier than computing both factorials separately.
When a permutation problem asks you to choose and arrange $k$ items from $n$ items, use the permutation formula $P(n,k) = \frac{n!}{(n-k)!}$ instead of computing full factorials manually.
Factorial is only defined for non-negative integers: expressions like $(-5)!$ or $(2.5)!$ have no meaning in elementary algebra.
For small factorials, write out the multiplication such as $5! = 5 \times 4 \times 3 \times 2 \times 1$ to check your arithmetic rather than relying on a calculator alone.

Related formulas

Sigma NotationSigma notation compresses long sums into a compact formula, letting you write and calculate the sum of many terms efficiently and accurately.Order of OperationsOrder of Operations is the set of rules that tells you which calculation to do first when you have multiple operations in one expression.Prime FactorizationPrime factorization breaks any whole number into a unique multiplication of primes—essential for finding common factors and simplifying fractions.

Frequently asked questions

Why is 0! = 1?: Zero factorial is defined as 1 by convention. While it might seem strange, this definition makes permutation and combination formulas work correctly. For instance, there is exactly one way to arrange zero objects (by doing nothing), which aligns with the definition $0! = 1$ .
What is the fastest way to calculate a large factorial?: Instead of multiplying all the numbers, look for opportunities to cancel. For example, $\frac{10!}{8!} = 10 \times 9$ rather than computing the full factorials. For factorials larger than about $12!$ , a calculator or computer is the practical choice since the results exceed millions.
Can you use factorial on negative numbers or decimals?: No; in elementary algebra, factorial is only defined for non-negative integers (0, 1, 2, 3, …). Expressions like $(-3)!$ or $(2.5)!$ have no meaning in this context. Advanced mathematics does define factorial for other values, but that is beyond Algebra II.
When do I need to use factorial in real life?: Factorials appear whenever you count arrangements or orderings: seating arrangements for events, scheduling tasks, arranging objects in a display, and many probability and statistics problems. Any time you ask 'how many different orders are possible?', factorial is likely the tool you need.

Need a different method?

The full solver is not scoped to one formula — type any problem and it will pick the method.

Open the math solver

Reviewed 2026-09-18