Factorial Notation

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FACTORIAL NOTATION

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Recall from the previous section ( Choosing Things: Does Order Matter?)
that Ckn denotes the number of ways to choose k items from n (without replacement) when the order DOESN'T matter;
it is the number of combinations that are possible when choosing k items from n items.

Also, Pkn denotes the number of ways to choose k items from n (without replacement) when the order DOES matter;
it is the number of permutations that are possible when arranging k items that are chosen from n items.

You will see in the next section that the formulas for Ckn and Pkn involve patterns like this:

10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
In this pattern, you start with a whole number, multiply by one less than this number, and continue until you get down to 1.
It is tedious and inefficient to write down this pattern, particularly when the starting number is large:

30 · 29 · 28 · 27 · 26 · 25 · 24 · 23 · 22 · 21 · 20 · 19 · 18 · 17 · 16 · 15 · 14 · 13 · 12 · 11 · 10 · 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1
Fortunately, there is a compact notation for this pattern, called factorial notation.
We can write:
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
The expression 10! is read aloud as ten factorial.

Remember that multiplication can be denoted in several ways.
In the examples below, the preferred (simplest) way is shown first:
3⋅2= 3×2=(3 )(2)
ab =a⋅b=a ×b=(a )(b)
In the definition below, the symbol := (an equal sign, with colon in front of it) emphasizes equal, by definition.

DEFINITION: FACTORIAL NOTATION
Let n be a whole number; i.e., a member of the set {0,1 ,2,3,... } .

For n>1 :

n!  :=  n(n-1) (n-2)× ...×1 Also:

1!  :=  1 and 0!  :=  1

The symbol n! is read aloud as n factorial .

Another pattern that frequently arises in counting arguments is something like this:

30 × 29 × 28 × 27 × 26 × 25 × 24
This pattern starts out the same as the pattern for 30! (30 factorial), but you don't continue all the way down to the number 1.
Clearly, this pattern can also be tedious and inefficient.

Is there a way to use factorial notation to get a simpler name for this kind of pattern? Yes!
The key is to multiply by 1 in an appropriate form, as shown below:

30⋅29⋅ 28⋅27⋅26 ⋅25⋅24
=30⋅29 ⋅28⋅27⋅ 26⋅25⋅24 ⋅23! 23!	(multiply by one in an appropriate form)
=30⋅ 29⋅28⋅27 ⋅26⋅25⋅ 24⋅23! 23!	(move numbers into the numerator)
=30! 23!	(Now, we have 30 down to 1 in the numerator)

This web exercise will give you practice with factorial notation.

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.