Permutations Examples
1. How many different starting rotations could
you make with 6 volleyball players?
(Positioning matters in a rotation.)
Permutations Examples
1. How many different starting rotations could
you make with 6 volleyball players?
(Positioning matters in a rotation.)
6•5•4•3•2•1 = 6! = 720
There are 6 options for the 1st position, then
5 options remaining for the 2nd position, 4
for the 3rd position, etc., until there is only 1
option left for the last position.
Permutations Examples
2. How many different starting lineups could
you make with 11 soccer players, if each
player could play any position?
Permutations Examples
2. How many different starting lineups could
you make with 11 soccer players, if each
player could play any position?
11! = 39,916,800
There are 11 options for the 1st position, then
10 options remaining for the 2nd position, 9
for the 3rd position, etc., until there is only 1
option left for the last position.
Permutations Examples
3. How many different starting lineups could
you make with 11 soccer players, if only 1
player can play goalie, 5 players can play any
of 5 forward positions, and 5 players can play
any of 5 defense/midfield positions?
Permutations Examples
3. How many different starting lineups could
you make with 11 soccer players, if only 1
player can play goalie, 5 players can play any
of 5 forward positions, and 5 players can play
any of 5 defense/midfield positions?
1•5!•5! = 14,400
Only 1 player can play goalie. For the
forwards, there are 5 options for the 1st
position, 4 options for the 2nd, etc. It works
the same for the 5 defenders.
Permutations Examples
4. How many seating charts could a teacher
make with 18 students in a class, and 18
available desks?
Permutations Examples
4. How many seating charts could a teacher
make with 18 students in a class, and 18
available desks?
18! = 6.4•1015
There are 18 options for the 1st seat, then
17 options remaining for the 2nd seat, 16 for
the 3rd seat, etc., until there is only 1 option
left for the last seat.
Permutations Examples
5. How many seating charts could a teacher
make with 18 students in a class, and 22
available desks?
Permutations Examples
5. How many seating charts could a teacher
make with 18 students in a class, and 22
available desks?
22•21•20•19•…•7•6•5 = 22! / (4!) = 4.68•1019
There are 22 seats to choose from for the 1st
student, then 21 seats remaining for the 2nd
student, 20 for the 3rd student, etc., until there are 5
seats left to choose from for the last student.
Permutations Examples
6. How many codes are possible for a lock
that has 4 digits, and each digit can be a
number 0-9?
Permutations Examples
6. How many codes are possible for a lock
that has 4 digits, and each digit can be a
number 0-9?
10•10•10•10 = 104 = 10,000
You can repeat numbers, so each digit has
10 possibilities (0-9).
Permutations Examples
7. How many codes are possible for a lock
that has 4 digits, and each digit can be a
number 0-9 or a letter A-F?
Permutations Examples
7. How many codes are possible for a lock
that has 4 digits, and each digit can be a
number 0-9 or a letter A-F?
16•16•16•16 = 164 = 65,536
The numbers 0-9 and the letters A-F form the
hexadecimal system, which is frequently used
with computers. As the name suggests there
are 16 possibilities for each digit.
Permutations Examples
8. In how many ways can you arrange 20
books on a bookshelf, if they are in a single
row?
Permutations Examples
8. In how many ways can you arrange 20
books on a bookshelf, if they are in a single
row?
20! = 2.43•1018
There are 20 books to choose from for the 1st
position, then 19 books remaining for the 2nd
position, 18 for the 3rd position, etc., until
there is only 1 book left for the last position.
Permutations Examples
9. In how many ways can you rank your
favorite 3 movies from a list of 10?
Permutations Examples
9. In how many ways can you rank your
favorite 3 movies from a list of 10?
10•9•8 = 720
The key word here is rank, indicating that
order matters. Ranking A-B-C as your first
three choices is different from ranking C-B-A
as your first three choices. Because order
matters, you do not need any division.
Permutations Examples
10. In how many ways can you rank your
favorite 5 books from a list of 20?
Permutations Examples
10. In how many ways can you rank your
favorite 5 books from a list of 20?
20•19•18•17•16
= 20! / (15!)
= P(20, 5) = 20 nPr 5
=1,860,480