FACTORIAL NOTATION

Recall from the previous section ( Choosing Things: Does Order Matter?)
that [beautiful math coming... please be patient] $\,{}_nC_k\,$ 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, [beautiful math coming... please be patient] $\,{}_nP_k\,$ 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 [beautiful math coming... please be patient] $\,{}_nC_k\,$ and $\,{}_nP_k\,$ involve patterns like this: [beautiful math coming... please be patient] $$ 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 $$ 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:
[beautiful math coming... please be patient] $$ 30 \cdot 29 \cdot 28 \cdot 27 \cdot 26 \cdot 25 \cdot 24 \cdot 23 \cdot 22 \cdot 21 \cdot 20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 $$ Fortunately, there is a compact notation for this pattern, called factorial notation.
We can write: [beautiful math coming... please be patient] $$ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 $$ 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: [beautiful math coming... please be patient] $$ \begin{gather} 3\cdot 2= 3\times 2 = (3)(2) \cr ab = a\cdot b=a \times b=(a)(b) \end{gather} $$

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 $\,\{0,1,2,3,\ldots\}\,$.

For [beautiful math coming... please be patient] $\,n\gt 1\,$: [beautiful math coming... please be patient] $$ n! \ :=\ n(n-1)(n-2) \times \cdots \times 1 $$ Also: [beautiful math coming... please be patient] $$ \begin{gather} 1!\ :=\ 1 \cr 0!\ :=\ 1 \end{gather} $$ The symbol ‘$\,n!\,$’ is read aloud as ‘$\,n\,$ factorial’.

Another pattern that frequently arises in counting arguments is something like this: [beautiful math coming... please be patient] $$ 30 \times 29 \times 28 \times 27 \times 26 \times 25 \times 24 $$ 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:

[beautiful math coming... please be patient] $30\cdot 29\cdot 28\cdot 27\cdot 26 \cdot 25\cdot 24$   
      [beautiful math coming... please be patient] $\displaystyle = 30\cdot 29 \cdot 28\cdot 27\cdot 26\cdot 25\cdot 24 \cdot \frac{23!}{23!}$ (multiply by one in an appropriate form)
      [beautiful math coming... please be patient] $\displaystyle = \frac{30\cdot 29\cdot 28\cdot 27 \cdot 26\cdot 25\cdot 24\cdot 23!}{23!}$ (move numbers into the numerator)
      [beautiful math coming... please be patient] $\displaystyle = \frac{30!}{23!}$(Now, we have $\,30\,$ down to $\,1\,$ in the numerator) 

This web exercise will give you practice with factorial notation.

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Combinations and Permutations


On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10
AVAILABLE MASTERED IN PROGRESS

(MAX is 10; there are 10 different problem types.)