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P Permutations versus Combinations
Contributed by: Cason

4! is read as 4 factorial

and is...

4 x 3 x 2 x 1.  It equals

24.

3! =

5! is read as 5 factorial

and is...

5 x 4 x 3 x 2 x 1.  It

equals 120.

7! = 7 x 6 x 5 x     

3! =         x        x 

0! is read as 0 factorial

and equals 1.

3! is read as 3 factorial

and is...

3 x 2 x 1.  It equals 6.

4! =       x      x     x

 
 
 
 

Permutations - order matters like combination locks

 

Of n things taken r at a time.

n

P

r

=           


or

n count down r times

(n-r)!

n!
 
 
 
 

N is the number of items being arranged in lots of r.

 

 

How many items are there to pick from           ?

n

P

r

=

           

or

n count down r times

(n-r)!

n!
 
 
 
 

N is the number of items being arranged in lots of r.

 

 

How many items are in each group?         items

n

P

r

=  

         

or

n count down r times

(n-r)!

n!
 
 

For this problem start at 6 and count down two place.

Permutations - order matters like combination locks  - 

Of n things taken r at a time.

n

P

r

6 x 5

n multiplied by count down r places

(n-r)!

n!

or

6

P

2

(6-2)!
6!
 
 

Permutations - order matters like combination locks  - 

Of n things taken r at a time.

n

P

r

6 x 5 x 4 =

=   

               or

n count down r times

(n-r)!

n!

6

P

3

(6-3)!
6!
 
 

Permutations - order matters like combination locks  - 

Of n things taken r at a time.

n

P

r

7 x 6 x 5 =

=  

               or

n count down r times

(n-r)!

n!

7

P

3

(7-3)!
7!
 
 

Permutations - order matters like combination locks  - 

Of n things taken r at a time.

5

P

2

or 5 x 4 = 20

(5-2)!
5!

5!
!
 
 
 

Permutations - order matters like combination locks  - 

Of n things taken r at a time.

8

P

2

count down 2 spots

8 x 7 =

(8-2)!
8!

8!
!
 
 
 

Permutations - order matters like combination locks  - 

Of n things taken r at a time.

8

P

3

   x     x      =  336

count down 3 spots
(8-3)!
8!

8!
!
 
 
 

Permutations - order matters like combination locks  - 

Of n things taken r at a time.

8

P

4

   x     x     x    =  1680

count down 4 spots
(8-4)!
8!

8!
!
 
 
 
 
 
 

Combinations - order does not matter - like ordering

a pizza with multiple toppings - of n things taken r at

a time.

n

n count down r times

C

r

=

r!

(n-r)!

n!
r!

or

 
 
 
 
 
 

Combinations - order does not matter - like ordering

a pizza with multiple toppings - of n things taken r at

a time.

n

n count down r times

   would be like the number of toppings to pick from

C

r

=

r!

(n-r)!

n!
r!

or

 
 
 
 
 
 

Combinations - order does not matter - like ordering

a pizza with multiple toppings - of n things taken r at

a time.

n

n count down r times

C

  is like the number of toppings you can select

r

=

r!

(n-r)!

n!
r!

or

 
 
 
 
 
 
 
 
 
 
 
 

Combinations - order does not matter - like ordering

a pizza with multiple toppings - of n things taken r at

a time.

or

n count down r times

6

C

2

=

r!
2!

(6-2)!

6!

=

(2•1)

6•5•4•3•2•1

3

6 x 5

2 x 1

(4•3•2•1)

=

 
 
 
 
 
 
 
 

Combinations - order does not matter - like ordering

a pizza with multiple toppings - of n things taken r at

a time.

or

n count down r times

6

C

3

=

r!
3!

(6-3)!

6!

=

6 x 5 x 4

3 x 2 x 1

(3•2•1)

6•5•4•3•2•1

(3•2•1)

=

 
 
 
 
 
 
 
 

Combinations - order does not matter - like ordering

a pizza with multiple toppings - of n things taken r at

a time.

or

n count down r times

7

C

3

=

r!
3!

(7-3)!

7!

=

(3•2•1)

7•6•5•4•3•2•1

7 x 6 x 5

3 x 2 x 1

(4•3•2•1)

=

 
 
 
 
 

Combinations - order does not matter - like ordering

a pizza with multiple toppings - of n things taken r at

a time.

5

C

2

=

2!

(5-2)!

5!

or

n count down r times

r!
 
 
 
 
 

Combinations - order does not matter - like ordering

a pizza with multiple toppings - of n things taken r at

a time.

8

C

2

=

2!

(8-2)!

8!

or

n count down r times

r!
 
 
 
 
 

Combinations - order does not matter - like ordering

a pizza with multiple toppings - of n things taken r at

a time.

5

C

3

=

3!

(5-3)!

5!

or

n count down r times

r!

Where a permutation of 3 things taken in lots of 3 has

6 choices (see choices below) because order matters,

a combination of 3 things taken in lots of 3 has

        choice(s) because order does not matter.

ABC     ACB

BAC     BCA

CAB     CBA

Possible choices for a   P   
3
3

Where a permutation of 3 things taken in lots of 2 has

6 choices (see choices below) because order matters,

a combination of 3 things taken in lots of 2 has

        choice(s) because order does not matter.

AB         AC

BA         BC

CA         CB

Possible choices for a   P   
3

2

Where a combination of 4 things taken in lots of 2 has

6 choice(s) because order does not matter a

permutation will have more choices as when the order

changes it makes a new set.  The permutation of 4

things taken in lots of 2 will have          choices.

1) AB = BA

2) AC = CA

3) AD = DA

Possible choices for a   C   

4) BC = CB

5) BD = DB

6) CD = CD

4

2

Runners finishing a race

Order matters as in (AB ≠ BA)  Permutation

Order doesn't matter as in (AB = BA) Combination

Four people applying for the same two jobs

Order matters as in (AB ≠ BA)  Permutation

Order doesn't matter as in (AB = BA) Combination

Four people applying for two different jobs

Order matters as in (AB ≠ BA)  Permutation

Order doesn't matter as in (AB = BA) Combination

Your locker combination

Order matters as in (AB ≠ BA)  Permutation

Order doesn't matter as in (AB = BA) Combination

Ordering pizza with several toppings

Order matters as in (AB ≠ BA)  Permutation

Order doesn't matter as in (AB = BA) Combination

Applying for the same jobs on a year book staff

Order matters as in (AB ≠ BA)  Permutation

Order doesn't matter as in (AB = BA) Combination

Four people applying for different positions on the

yearbook staff

Order matters as in (AB ≠ BA)  Permutation

Order doesn't matter as in (AB = BA) Combination

Running for office - first place votes is President,

second place, Vice President, etc.

Order matters as in (AB ≠ BA)  Permutation

Order doesn't matter as in (AB = BA) Combination

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