Key takeaway: A factorial n! is the product of all whole numbers from n down to 1; it counts the ways to arrange n different objects in a row, and 0! is defined as 1.
Grade 12 – PrecalculusS-CP.B.9Probability and Counting
Definition of Factorial
A factorial multiplies a whole number by every positive whole number below it. Written with an exclamation mark, \(4!\) is the product of 4, 3, 2 and 1, which is 24.
The operation answers a common counting question: in how many orders can \(n\) different things be lined up? There are \(n\) choices for the first spot, then \(n-1\) for the second, and so on down to 1. The values grow remarkably fast; 10! already tops three million. By agreement, \(0!=1\), which keeps counting formulas consistent when no objects are being arranged.
Formula and notation
For a whole number \(n\ge 1\):
\[n!=n\,(n-1)\,(n-2)\cdots 3\cdot 2\cdot 1,\qquad n!=n\cdot(n-1)!,\qquad 0!=1.\]
The recursive form \(n!=n\cdot(n-1)!\) is the key to simplifying fractions of factorials.
Key facts
- Only whole numbers zero or larger qualify in this course.
- The values for 1 through 6 are 1, 2, 6, 24, 120 and 720.
- Cancel the shared tail when dividing: 12!/9! leaves just 12, 11 and 10 multiplied.
- The operation is not distributive: (a + b)! differs from a! + b!.
Factorial example
Six friends line up for a group photo. How many different orders are possible? Then simplify \(\frac{12!}{9!}\).
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Six choices for the first spot, five for the next, and so on: \(6!=720\).
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Write the top as \(12\cdot 11\cdot 10\cdot 9!\) so that \(9!\) cancels with the bottom.
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What remains: \(12\cdot 11\cdot 10=1320\).
There are 720 orders, and \(\frac{12!}{9!}=\)1,320.
Common mistake
Watch out
Rhea claims \(3!\cdot 2!=5!\), since \(3+2=5\).
Why it is wrong
Compute each side: \(3!\cdot 2!=6\cdot 2=12\), while \(5!=120\). A factorial of a sum is not a product of factorials, and this operation does not distribute over addition.
Related terms and lessons
Browse the glossary: A–Z list · Precalculus vocabulary · Probability and Counting