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.
\[
\underline{5} * \underline{4} * \underline{3} * \underline{2} * \underline{1}=120 \text { choices }
\]
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 \(\underline{7} * \underline{6} * \underline{5}=210\) 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 \(\underline{7} * \underline{6} * \underline{5}\) would be represented as \(_{7} P_{3}\) or
\[
P(7,3)
\]

The standard definition of this notation is:
\[
_{n} P_{r}=\frac{n !}{(n-r) !}
\]
You can see that, in the example, we were interested in \(_{7} P_{3},\) which would be calculated as:
\[
_{7} P_{3}=\frac{7 !}{(7-3) !}=\frac{7 !}{4 !}=\frac{7 * 6 * 5 * 4 * 3 * 2 * 1}{4 * 3 * 2 * 1}
\]
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:
\[
_{7} P_{3}=7 * 6 * 5=210
\]
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 \(\quad\) 2nd place: Charlie 3rd place: Charlie \(\quad\) 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 \(_{5} P_{5}\)
\[
_{5} P_{5}=\frac{5 !}{(5-5) !}=\frac{5 !}{0 !}=\frac{120}{1}=120
\]
This is the reason why \(0 !\) is defined as 1

EXERCISES 7.2
1) \(\quad 4 * 5 !\)
2) \(\quad 3 ! * 4 !\)
3) \(\quad 5 ! * 3 !\)
4) \(\quad \frac{8 !}{6 !}\)
5) \(\quad \frac{10 !}{7 !}\)
6) \(\quad \frac{9 ! * 6 !}{3 ! * 7 !}\)
7) \(\quad \frac{12 ! * 3 !}{8 ! * 6 !}\)
8)\(\quad_{10} P_{4}\)
9) \(\quad_{4} P_{3}\)
10) \(\quad_{7} P_{5}\)
11) \(\quad_{9} P_{2}\)
12) \(\quad_{8} P_{4}\)
13) \(\quad\) so \(P_{3}\)
14) \(\quad n_{1}\)
15) \(\quad_{10} P_{r}\)
16) List all the permutations of the letters \(\{a, b, c\}\)
17) List all the permutations of the letters \(\{a, b, c\}\) taken two at a time.
18) How many permutations are there of the group of letters \(\{a, b, c, d, e\} ?\)
19) How many permutations are there of the group of letters \(\{a, b, c, d\} ?\)

List these permutations.
20) How many ways can a president, vice president and secretary be chosen from a group of 20 students?
21) How many ways can a president, vice president, secretary and treasurer be chosen from a group of 50 students?
22) How many ways can 5 boys and 5 girls be seated in a row containing ten seats:
\(\quad\) a) with no restrictions?
\(\quad\) b) if boys and girls must alternate seats?
23) How many ways can 5 boys and 4 girls be seated in a row containing nine seats:
\(\quad\) a) with no restrictions?
\(\quad\) b) if boys and girls must alternate seats?
24) How many ways can 6 people be seated if there are 10 chairs to choose from?
25) How many ways can 4 people be seated if there are 9 chairs to choose from?
26) How many ways can a group of 8 people be seated in a row of 8 seats if two people insist on sitting together?
27) How many ways can a group of 10 people be seated in a row of 10 seats if three people insist on sitting together?