1. 3.7 Counting Techniques: Permutations

2. If we line up all the students in this class, what is the probability that Aninidita and Hanine are standing together?
If I randomly create groups of 3, what is the probability that Marian, Ngoc, and Sumi are
all in the same group?
all in different groups?
These require techniques to count the outcomes of more complex experiments
combinatorics: permutations and combinations

3. How many different ways can we line up the students in this class?
Let’s try it! (or at least, let’s try a small sample…)

4. Counting Techniques
Example: Placing 3 objects, A, B, C in line for a game
Resulting Order
First
Second
Third B
A BC C A
ACB C B A
BAC B C C A
BCA C B A
CAB B A
CBA
# different orders possible =
Multiplicative = x x
= 3! 6

5. Factorial Notation n! = n(n-1)(n-2)…(3)(2)(1) 4! = 4(3)(2)(1) = 24
n! represents the number of ways n different objects can be ordered
that is, the number of ways n objects can be selected to create ordered arrangements of size n

6. Example 1
How many ways can we arrange Andrey, Bibi, Charles, Deb, and Eunice in a line-up?
Note: you can use the x! button on your calculator!

7. Permutation P(n, r) = nPr
A ordered arrangement of n different objects taken r at a time
(n! means we arrange all n objects; a permutation only arranges r of those n objects)
The total number of possible arrangements, or permutations, of r objects taken from a set of size n is denoted by
P(n, r) = nPr

8. How? How many ways can we arrange 3 objects from a group of 6? 6 5 4
We write:

9. Example 3
How many ways can we arrange 4 books out of 10?

10. Example 4 nPr You can also use the nPr button on your calculator.
How many ways can we arrange 30 objects out of 100?
Error on the calculator!
nPr
Type: 100
30 =

11. Permutations with Identical Elements
Consider DOL1L2
# permutations = 4! = 24
Examples: DOL1L2 DOL2L1
DL1L2O DL2L1O
L1L2OD L2L1OD
DOLL DOLL
DLLO DLLO
LLOD LLOD
The L’s are the same
Out of 24 arrangements, only 12 are different from each other
The number of different arrangements is

12. Permutations with Identical Elements
The number of permutations of a set of n objects containing a identical objects of one kind, b identical objects of a second kind, c identical objects of a third kind, etc., is