College Algebra & Trigonometry, 2018a

7.2. FACTORIAL NOTATION AND PERMUTATIONS 313
The standard definition of this notation is:
nP r =
n!
(n − r)!
You can see that, in the example, we were interested in 7 P 3 , which would be
calculated as:
7P 3 =
7!
(7 − 3)! = 7!
4! = 7 ∗ 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1
4 ∗ 3 ∗ 2 ∗ 1
The 4 ∗ 3 ∗ 2 ∗ 1 in the numerator and denominator cancel each other out, so we
are just left with the expression we fouind intuitively:
7P 3 =7∗ 6 ∗ 5 = 210
Although the formal notation may seem cumbersome when compared to the intuitive
solution, it is handy when working with more complex problems, problems
that involve large numbers, or problems that involve variables.
Note that, in this example, the order of finishing the race is important. That is to
say that the same three contestants might comprise different finish orders.
1st place: Alice 1st place: Bob
2nd place: Bob 2nd place: Charlie
3rd place: Charlie 3rd place: Alice
The two finishes listed above are distinct choices and are counted separately in
the 210 possibilities. If we were only concerned with selecting 3 people from a
group of 7, then the order of the people wouldn’t be important - this is generally
referred to a “combination” rather than a permutation and will be discussed in the
next section.
Returning to the original example in this section - how many different ways are
there to seat 5 people in a row of 5 chairs? If we use the standard definition of
permutations, then this would be 5 P 5 .
5P 5 =
This is the reason why 0! is defined as 1.
5!
(5 − 5)! = 5!
0! = 120
1 = 120