Section 6.4 – Permutations and Combinations
1
Section 6.4
Permutations and Combinations
Definition:
nFactorial
For any natural number
n
,
1
2
3
)
2
)(
1
(
!
⋅
⋅
⋅
⋅
⋅


=
n
n
n
n
.
Note: 0! = 1
Example:
2
1
.
2
!
2
=
=
;
120
1
.
2
.
3
.
4
.
5
!
5
=
=
.
Arrangements:
n
different objects can be arranged in
!
n
different ways.
Example:
In how many different ways can you arrange 4 different books on a shelf?
Example:
In how many ways can 5 people be seated?
Arrangements of n objects, not all distinct:
Given a set of n objects in which
1
n
objects are alike and of one kind,
2
n
objects are alike and of
another kind,…, and, finally,
r
n
objects are alike and of yet another kind so that
n
n
n
n
r
=
+
+
+
...
2
1
, then the these objects can be arranged in:
!
!
!
!
2
1
r
n
n
n
n
⋅
⋅
⋅
different ways.
Example:
In how many ways can you arrange 2 red, 4 blue and 5 black pencils?
Example:
How many 4 letter words can you form by arranging the letters of the word MATH?
Example:
How many 8 letter words can you form by arranging the letters of the word
MINIMIZE?
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Section 6.4 – Permutations and Combinations
2
Definition:
A
permutation
is an arrangement of a specific set where the order in which the
objects are arranged is important.
Formula:
P
(
n
,
r
) =
)!
(
!
r
n
n

,
r <
n
where
n
is the number of distinct objects and
r
is the number of distinct objects taken
r
at a time.
Example:
How many 2digit numbers can be written by using the digits 1,2 and 3? (Repetition is
not allowed)
E= {1, 2, 3}
List: 12, 13, 21, 23, 31, 32
6
1
6
!
1
!
3
)!
2
3
(
!
3
)
2
,
3
(
=
=
=

=
P