Factorials and the Gamma Function
The notion of factorials pops up in many branches of mathematics and in many
applications. The factorial of a whole number is defined as follows.
n! = 1⋅ 2⋅ 3L(n -1) ⋅ n
For example;
3! = 6
10! = 3628800
There is a button on your calculator to do this for you. It is under the math menu, then
under the prob menu. There are two related buttons, nPr and nCr.
n! = number of different orders (or arrangements) of n objects.
For example, ABC can be arranged in 3! different ways. ABC, ACB, BCA, BAC, CAB,
CBA. 3! = 6.
nPr = the number of different permutations of n objects taken r at a time.
nPr =
n!
(n - r)!
For example consider the 4 objects ABCD, how many different 2 letter objects can be
made? AB, AC, AD, BC, BD, CD, BA, CA, DA, CB, DB, DC.
4
P2 =
4!
4!
= = 4 ⋅ 3 = 12
(4 - 2)! 2!
nCr = the number of different combinations of n objects taken r at a time.
nCr =
n!
r!(n - r)!
This is different from permutation in that it is now not important which order the objects
appear. Like when you are playing cards, if you are dealt a jack it does not matter if you
get it first or second or third. You still have a jack. Permutations are more like when we
spell. The order of the letters is very important.
For example, how many different 5 card hands are there from a 52 card deck?
C5 =
52
52!
52!
=
= 2598960
5!(52 - 5)! 5!47!
This is a number you should remember the next time you try to draw to an inside straight.
Long ago it was found necessary to generalize the factorial function to allow for nonwhole numbers. e.g. (2.33)! This generalized object was soon found in all sorts of
applications from science to engineering. Since it occurred so often it got its very own
name, the Gamma function. Once you have used it a few times you may simply refer to
it as Gamma. There are many equations that define the Gamma function, some of the
most common are given below.
G(n) =
Ú
•
0
x n-1e -x dx
n > 0
k!
kn
k Æ• (n )(n + 1)L(n + k )
G(n) = lim
e-gx
G(x) =
•
x ’((1+ x /m)e -x / m )
m =1
g is Euler’s constant. Its numerical value is 0.5772156649015328606 . . . It is a
transcendental number, one of those numbers that goes on forever and does not repeat.
Those formulae look pretty messy and hard to use. Fear not! You will probably never
have to use them. Your calculator knows how to compute values for the Gamma
function. You simply use the factorial button. Below are some properties that will help
you for the numbers you might need.
G( n + 1) = nG(n )
G(n + 1) = n! if n is a whole number.
G(1/2) = p
G(n) = undefined, that is, ± • if n is a negative integer or zero. You might want to
make a graph to see why.
G(m + 1/2) =
G(1/2 - m) =
1⋅ 3⋅ 5L(2m -1)
p
2m
(-1)m 2m
1⋅ 3⋅ 5L(2m - 1)
p
The vertical (or y axis) is scaled by 10. 1 on the y axis is actually 10.
The reciprocal of the gamma function is also encountered frequently in applications.
Note the location of the zeros of the function.