Factorial Function

If \(n\) is a positive integer, then we define \(n!\) (pronounced \(n\) factorial) to be the product of integers: \(n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1\).

For example,

• \(3! = 3 \times 2 \times 1 = 6\)
• \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)
• \(2! = 2 \times 1 = 2\)
• \(1!\) = 1

Calculating Factorials

Let's have a look at the patterns that factorials make:

So we can find the factorial of a positive integer \(n\) by multiplying \(n\) by the factorial of the integer that is one less than \(n\):

This is an example of a recursive function definition: a function that is defined in terms of its earlier values.

For a recursive function definition to make sense, we have to specify a starting value. For the factorial function, we have defined the starting value to be

The factorials of negative integers are not defined.

• If \(10! = 3,628,800\), what is the value of \(11!\)?
  Solution:
  \(11! = 11 \times 10! = 11 \times 3,628,800 = 39,916,800\)
• If \(7! = 5,040\), what is the value of \(8!\)?
  Solution:
  \(8! = 8 \times 7! = 8 \times 5,040 = 40,320\)
• \(1,236! = 1,236 \times 1,235!\), but I won't write this down as it is HUGE!

Calculations with Factorials

We can use the recursive definition of the factorial function given above to simplify calculations with factorials. This is particularly important when we are calculating values for combinations such as \(\begin{pmatrix} n \\ r \end{pmatrix} = \dfrac{n!}{r!(n - r)!}\).

Example:

Find \(\dfrac{6!}{3!}\).

Solution:

You can either calculate:

or you can use the recursive definition to more quickly calculate

Example:

Find \(\dfrac{153!}{151!}\)

Solution:

This time, I definitely don't want to write down the full expansions for \(153!\) and \(151!\) and then cancel. That would take forever! I'll stick to the recursive definition:

Interesting Facts About the Factorial Function

The factorial function gets really big, really, very, super fast. It grows more quickly than \(x^n\) for any \(n\), or any exponential function like \(2^n\), \(e^n\) or \(1,000^n\). For example, \(70!\) is larger than a Googol (\(10^{100}\)).

There are \(28!\) ways to arrange a set of double-six dominoes in a line, and \(55!\) ways to arrange a set of double-nine dominoes in a line.

There are about \(23!\) stars in the universe.

Statisticians are interested in factorials of fractional numbers such as \(\dfrac{1}{2}\). These are defined using Gamma functions. The Gamma functions of negative integers are undefined, but we can find the values of Gamma functions for fractional values such as \(\dfrac{1}{2}\). Statisticians consider this to be the factorial of \(\dfrac{1}{2}\), and its value is \(\frac{1}{2}! = \frac{1}{2}\;\sqrt{\pi}\). We can use this value to calculate some other half-integer factorials such as \(\frac{7}{2}!\) and \(-\frac{11}{2}!\) using the recursive defintion of the factorial function. For example,

See if you can work out some other half-integer factorials.