College Algebra & Trigonometry, 2018a

312 CHAPTER 7. COMBINATORICS
7.2 Factorial Notation and Permutations
In considering the number of possibilities of various events, particular scenarios
typically emerge in different problems. One of these scenarios is the multiplication
of consecutive whole numbers. For example, given the question of how
many ways there are to seat a given number of people in a row of chairs, there
will obviously not be repetition of the individuals. So, if we wanted to know how
many different ways there are to seat 5 people in a row of five chairs, there would
be 5 choices for the first seat, 4 choices for the second seat, 3 choices for the third
seat and so on.
5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 = 120 choices
In these situations the 1 is sometimes omitted because it doesn’t change the value
of the answer. This process of multiplying consecutive decreasing whole numbers
is called a “factorial.” The notation for a factorial is an exclamation point. So
the problem above could be answered: 5! = 120. By definition, 0! = 1. Although
this may not seem logical intuitively, the definition is based on its application in
permutation problems.
A“permutation” uses factorials for solving situations in which not all of the possibilities
will be selected.
So, for example, if we wanted to know how many ways can first, second and third
place finishes occur in a race with 7 contestants, there would be seven possibilities
for first place, then six choices for second place, then five choices for third place.
So, there are 7 ∗ 6 ∗ 5 = 210 possible ways to accomplish this.
The standard notation for this type of permutation is generally n P r or P (n, r).
This notation represents the number of ways of allocating r distinct elements into
separate positions from a group of n possibilities.
In the example above the expression 7 ∗ 6 ∗ 5 would be represented as 7 P 3 or
P (7, 3).