What a factorial is and why you need it

A factorial is the product of a whole number and every whole number below it, down to 1. The symbol is an exclamation mark. So 5 factorial, written as 5!, means 5 × 4 × 3 × 2 × 1, which equals 120. Factorials appear in probability, statistics, and combinatorics — anywhere you need to count arrangements or combinations of things.

You will encounter factorials in high school algebra, college mathematics, and some professional fields like data analysis. The computation itself is straightforward once you know the pattern, but the numbers grow very large very quickly, which is why calculators become necessary for anything beyond small numbers.

Key Takeaways

  • A factorial multiplies a number by every positive whole number below it: 4! = 4 × 3 × 2 × 1 = 24.
  • By definition, 0! and 1! both equal 1, which is a rule you must memorize.
  • You can compute factorials by hand for numbers up to about 10, after which a scientific calculator becomes practical.
  • Most graphing calculators and scientific calculators have a factorial button (usually labeled n! or x!) that computes the result when ready.
  • Factorials grow so fast that 20! is already over 2 trillion, making hand calculation impractical beyond the teens.

Computing factorials by hand for small numbers

Start with the number you want the factorial of. Write it down, then multiply it by the next smaller number, then the next, until you reach 1. For example, to find 6!:

6! = 6 × 5 × 4 × 3 × 2 × 1

Work left to right. First, 6 × 5 = 30. Then 30 × 4 = 120. Then 120 × 3 = 360. Then 360 × 2 = 720. Then 720 × 1 = 720. So 6! = 720.

This method works well for 2! through 10!. Beyond that, the numbers become large enough that mistakes in multiplication become likely, and a calculator saves time and reduces error.

The special cases: 0! and 1!

By definition, 0! = 1 and 1! = 1. These are not computed; they are rules. The reason 0! equals 1 comes from combinatorics — there is exactly one way to arrange zero objects — but you do not need to understand the reasoning. You only need to remember the rule.

When you see a factorial problem that involves 0 or 1, substitute 1 when ready. If you are asked to compute 5! ÷ 3!, you can simplify by canceling: (5 × 4 × 3 × 2 × 1) ÷ (3 × 2 × 1) = 5 × 4 = 20. Recognizing that 3! appears in both the numerator and denominator lets you skip the full calculation.

Using a scientific calculator

Most scientific calculators have a factorial button. Look for a key labeled n!, x!, or sometimes ! on its own. On some models it is a secondary function accessed by pressing Shift or 2nd first.

To compute 8! on a typical scientific calculator: enter 8, then press the n! button. The display shows 40320. That is the answer. No multiplication steps needed.

If your calculator does not have a factorial button, you can still compute it by hand using the multiplication method above, or you can use an online calculator. Most graphing calculators (like the TI-84) also have a factorial function, usually found in the Math menu under PRB (Probability).

Using an online calculator or spreadsheet

If you do not have a scientific calculator nearby, you can compute factorials using a web browser. Search "factorial calculator" and you will find several free tools that let you enter a number and when ready see the result.

You can also use a spreadsheet program like Microsoft Excel or Google Sheets. In Excel, the function is =FACT(n), where n is the number. Type =FACT(5) in a cell and press Enter, and the spreadsheet returns 120. Google Sheets uses the same syntax.

These tools are useful when you need to compute large factorials — anything above 20 — because the numbers become so large that even calculators may display them in scientific notation (like 2.43E+18 instead of the full number).

Why factorials grow so fast

Each time you increase the number by 1, you multiply the previous factorial by that new number. So 5! = 120, and 6! = 120 × 6 = 720. This means factorials double or triple or more with each step up. By 10!, you are already at 3,628,800. By 20!, the number exceeds 2 trillion.

This rapid growth is why hand calculation becomes impractical quickly. It is also why factorials appear in problems about probability and combinations — the numbers represent how many different ways things can be arranged, and that count explodes as the set gets larger.

Common mistakes to avoid

The most common error is forgetting that 0! = 1. Students often assume 0! = 0, which is wrong. Remember: 0! is defined as 1, period.

Another mistake is stopping the multiplication too early. When you compute 7!, you must go all the way down to 1. Stopping at 2 and forgetting the final multiplication by 1 will not change your answer (since multiplying by 1 does nothing), but stopping at 3 or higher will give you the wrong result.

If you are using a calculator, make sure you are pressing the factorial button and not a different function. On some calculators, the factorial button is near the permutation (nPr) or combination (nCr) buttons, which do different things entirely.

Frequently Asked Questions

Can you compute a factorial of a negative number?

No. Factorials are only defined for whole numbers 0 and above. If you see a problem asking for the factorial of a negative number, there is an error in the problem itself. Factorials represent counting arrangements, which only makes sense for zero or positive quantities.

What is the largest factorial a calculator can compute?

Most scientific calculators can compute factorials up to 69! or 70!. Beyond that, the number becomes too large for the calculator to store. If you need a factorial larger than that, use an online tool or a computer program, which can handle much bigger numbers.

Do I need to memorize small factorials?

It helps to know 0! = 1, 1! = 1, 2! = 2, 3! = 6, 4! = 24, and 5! = 120. These appear often enough in problems that memorizing them saves time. Beyond 5!, you can compute or look them up as needed.

Why is 0! equal to 1 and not 0?

The definition comes from combinatorics: there is exactly one way to arrange zero objects (the empty arrangement). Mathematicians defined 0! = 1 to make formulas work consistently. You do not need to understand the reasoning — just remember the rule.

What is the difference between factorial and permutation?

A factorial (n!) counts all possible arrangements of n objects. A permutation (nPr) counts arrangements of r objects chosen from a larger set of n objects. Permutations use factorials in their formula, but they are not the same thing. If a problem asks for permutations, use the nPr button on your calculator, not the factorial button.