4.3. GENERALIZED PERMUTATIONS AND COMBINATIONS
67
4.3. Generalized Permutations and Combinations
4.3.1. Permutations with Repeated Elements.
Assume that
we have an alphabet with
k
letters and we want to write all possible
words containing
n
1
times the first letter of the alphabet,
n
2
times the
second letter,...,
n
k
times the
k
th letter. How many words can we
write? We call this number
P
(
n
;
n
1
,n
2
,...,n
k
), where
n
=
n
1
+
n
2
+
· · ·
+
n
k
.
Example
: With 3
a
’s and 2
b
’s we can write the following 5letter
words:
aaabb
,
aabab
,
abaab
,
baaab
,
aabba
,
ababa
,
baaba
,
abbaa
,
babaa
,
bbaaa
.
We may solve this problem in the following way, as illustrated with
the example above. Let us distinguish the different copies of a letter
with subscripts:
a
1
a
2
a
3
b
1
b
2
. Next, generate each permutation of this
five elements by choosing 1) the position of each kind of letter, then 2)
the subscripts to place on the 3
a
’s, then 3) these subscripts to place on
the 2
b
’s. Task 1) can be performed in
P
(5;3
,
2) ways, task 2) can be
performed in 3! ways, task 3) can be performed in 2!. By the product
rule we have 5! =
P
(5;3
,
2)
×
3!
×
2!, hence
P
(5;3
,
2) = 5!
/
3!2!.
In general the formula is:
P
(
n
;
n
1
,n
2
,...,n
k
) =
n
!
n
1
!
n
2
!
... n
k
!
.
4.3.2. Combinations with Repetition.
Assume that we have a
set
A
with
n
elements. Any selection of
r
objects from
A
, where each
object can be selected more than once, is called a
combination of
n
objects taken
r
at a time with repetition
. For instance, the combinations
of the letters
a,b,c,d
taken 3 at a time with repetition are:
aaa
,
aab
,
aac
,
aad
,
abb
,
abc
,
abd
,
acc
,
acd
,
add
,
bbb
,
bbc
,
bbd
,
bcc
,
bcd
,
bdd
,
ccc
,
ccd
,
cdd
,
ddd
. Two combinations with repetition are considered identical
if they have the same elements repeated the same number of times,
regardless of their order.
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 Spring '10
 Hald,OH
 Differential Equations, Linear Algebra, Algebra, Combinatorics, Number Theory, Permutations, Equations, Permutations And Combinations, binomial coefficients

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