Factorial
In short: A factorial, written n!, multiplies a whole number by every smaller positive whole number, and the result counts the different line-ups possible for n distinct things.
PrecalculusProbabilityS-CP.B.9
What is Factorial?
A factorial is shorthand for a long countdown multiplication. Seven distinct pastries in a display row can be ordered \(7!\) ways: 7 candidates for spot one, 6 left for spot two, and so forth.
By definition \(n! = n(n-1)(n-2)\cdots 2\cdot 1\) for a positive integer \(n\), and \(0! = 1\) by agreement, so that formulas for counting still work. Values grow extremely fast: \(5! = 120\) but \(10! = 3{,}628{,}800\). Factorials also satisfy \(n! = n\cdot(n-1)!\).
How to spot it
- An exclamation mark follows a whole number or a variable, as in 6! or (n + 2)!.
- A problem asks in how many orders items can be arranged, lined up or scheduled.
- A counting formula for permutations or combinations contains n! over a product of other factorials.
- A fraction of factorials, such as 10!/(7!3!), must be simplified without a calculator.
The method
- Write out the product from \(n\) down to 1, or from \(n\) down to the point where cancellation starts.
- In a fraction, cancel the shared tail: \(\frac{10!}{7!} = 10\cdot 9\cdot 8\).
- Remember \(0! = 1\) and \(1! = 1\).
- Multiply what remains and keep an eye on very large totals, which may need a calculator.
Factorial: solved example
Problem
A café displays 7 different pastries in a row. Find the number of possible orders, and then compute \(\frac{10!}{7!\,3!}\).
Solution
The orders number \(7! = 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1 = 5{,}040\).
For the fraction, cancel \(7!\): \(\frac{10\cdot 9\cdot 8}{3\cdot 2\cdot 1} = \frac{720}{6} = 120\). The results are 5,040 and 120.
Try it
Question 1
What is \(0!\), and why is the answer useful?
Show the answer
\(0! = 1\). It lets counting formulas such as \(\frac{n!}{n!\,0!}\) equal 1 for choosing all or none of the items.
Question 2
Simplify \(\frac{(n+1)!}{(n-1)!}\).
Show the answer
\(\frac{(n+1)\,n\,(n-1)!}{(n-1)!} = (n+1)n\).
Factorial vs. Permutation
The factorial takes one number and counts line-ups of all n items. A permutation counts line-ups of only a chosen few, like 3 medal winners from 12 runners, which equals \(\frac{12!}{9!}\). Permutations are assembled from factorials, yet the two ideas differ.
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