Why is 0 factorial equal to 1?

Zero factorial equals 1 by definition, and that definition keeps the pattern unbroken. Walking down the list, you divide by the top number each time: 3! = 4! / 4 = 6, 2! = 3! / 3 = 2 and 1! = 2! / 2 = 1. One more step gives 0! = 1! / 1 = 1. The game itself never asks for 0!, because its smallest factorial is 1!.

How do you work out n factorial without a calculator?

Build n factorial from the one just below it, so each step costs a single multiplication: knowing 5! = 120 gives 6! = 6 × 120 = 720. For one big factorial, pair the numbers into easy products. In 8!, take 8 × 5 = 40, 7 × 6 = 42 and 4 × 3 × 2 = 24. Then 40 × 42 = 1680, and 1680 × 24 = 40320.

Is there a quick way to check a factorial answer?

Yes, the last digit gives you a fast check. From 5! on, every factorial contains both 5 and 2, so it ends in 0: 120, 720, 5040 and 40320. From 10! on, it ends in at least two zeros. If your 7! ends in any other digit, a multiplication slipped. For a quotient, multiply your answer by the bottom factorial, and you should get the top one.

Why does my child get factorial division wrong?

Most often, they divide the numbers in front of the signs, so 6! / 3! turns into 6 / 3 = 2. But each ! stands for a whole countdown, and the real answer is 720 / 6 = 120. Ask them to write both countdowns out, cross off the numbers they share and multiply what is left: 6 × 5 × 4 = 120. After a few rounds, the crossing out happens in their head.

Is this factorials game free for a class, and how would a lesson use it?

Yes: neither factorial level asks for money, an account or an install; students just open a browser. To start a lesson with factorials practice, write 10! / 8! on the board and ask for it without a calculator: cancelling leaves 10 × 9 = 90. Then try 100! / 98!, which looks impossible yet is only 100 × 99 = 9900. With the schools plan, billed per student, a teacher sees on one dashboard which students are improving.

Math practice · reimagined

Factorial Game

This factorial challenge runs from 1! = 1 to 8! = 40320 and mixes in quotients like 7! / 4!, where most of the countdown cancels out. Your rival gets the same problem, so every multiplication you skip lets your fighter swing sooner.

Ages 4–99+ · no download · plays in your browser · 23 languages

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What is a factorial?

A factorial multiplies a whole number by every whole number below it, down to 1. It is written with an exclamation mark, so 5! is read “five factorial” and means 5 × 4 × 3 × 2 × 1 = 120. Each new factorial multiplies the one before by one more number, which is why 10! already reaches 3628800.

Every level of Factorial

Example: what is 6! / 4!?

Write out both countdowns: 6! = 6 × 5 × 4 × 3 × 2 × 1 and 4! = 4 × 3 × 2 × 1.
The numbers multiplied in 4! also sit inside 6!, so they cancel out.
Only 6 × 5 is left on top, and 6 × 5 = 30.
The long way agrees, since 720 / 24 = 30, so you type 30.

Learn it step by step

The little "!" means multiply every whole number from your number down to 1.

4! = 4 × 3 × 2 × 1 = 24. It counts how many ways you can line things up: 4 different toys can be ordered in 24 different rows. Notice how fast it grows!

Multiply the countdown and type the result. Some problems divide one factorial by another, like 6! / 2! — work each countdown out, then divide.

Tips and common mistakes

Tie two values to the clock: 4! = 24 is the number of hours in a day, and 6! = 720 is the number of hours in 30 days.
A common slip is adding instead of multiplying: 5 + 4 + 3 + 2 + 1 = 15, but 5! = 120. Say “times” at each step.
Flip cards 1 to 8 in order while your teenager multiplies aloud, each card times the last result: 1, 2, 6, 24, up to 40320.

How to play

Each round shows a factorial such as 5! or a quotient such as 7! / 4!, and you type the whole-number answer on the keypad, up to five digits once 8! = 40320 appears. Only the first right answer lands a hit, so cancel before you multiply: the faster you get there, the harder Onion strikes.

— Quick answers —

Factorial: the short FAQ

Why is 0 factorial equal to 1?: Zero factorial equals 1 by definition, and that definition keeps the pattern unbroken. Walking down the list, you divide by the top number each time: 3! = 4! / 4 = 6, 2! = 3! / 3 = 2 and 1! = 2! / 2 = 1. One more step gives 0! = 1! / 1 = 1. The game itself never asks for 0!, because its smallest factorial is 1!.
How do you work out n factorial without a calculator?: Build n factorial from the one just below it, so each step costs a single multiplication: knowing 5! = 120 gives 6! = 6 × 120 = 720. For one big factorial, pair the numbers into easy products. In 8!, take 8 × 5 = 40, 7 × 6 = 42 and 4 × 3 × 2 = 24. Then 40 × 42 = 1680, and 1680 × 24 = 40320.
Is there a quick way to check a factorial answer?: Yes, the last digit gives you a fast check. From 5! on, every factorial contains both 5 and 2, so it ends in 0: 120, 720, 5040 and 40320. From 10! on, it ends in at least two zeros. If your 7! ends in any other digit, a multiplication slipped. For a quotient, multiply your answer by the bottom factorial, and you should get the top one.
Why does my child get factorial division wrong?: Most often, they divide the numbers in front of the signs, so 6! / 3! turns into 6 / 3 = 2. But each ! stands for a whole countdown, and the real answer is 720 / 6 = 120. Ask them to write both countdowns out, cross off the numbers they share and multiply what is left: 6 × 5 × 4 = 120. After a few rounds, the crossing out happens in their head.
Is this factorials game free for a class, and how would a lesson use it?: Yes: neither factorial level asks for money, an account or an install; students just open a browser. To start a lesson with factorials practice, write 10! / 8! on the board and ask for it without a calculator: cancelling leaves 10 × 9 = 90. Then try 100! / 98!, which looks impossible yet is only 100 × 99 = 9900. With the schools plan, billed per student, a teacher sees on one dashboard which students are improving.

More math to master

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