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