For permutations with
repetition, order still matters.
Hint:
When working with
"arrangements", it is often helpful to make a visual of
the situation by drawing segments to represent
the locations of the items.

Permutations with Repetition:
Example: How many 3 letter words can be
formed using the letters
c,a,t
allowing for repetition of the letters?
Solution:
For this problem, 3 locations are
needed:
_____ • _____ •
_____
There are 3 letters which can be used to fill
the first location. Because repetition is
allowed, the same 3 letters can
be used to fill the second location and also the
last location.
__3___ • __3___ •
__3___ = 27 arrangements 
27 arrangements
with repetition: 
ccc
caa
ctt
cat
cta
cac
ctc
cca
cct 
aaa
acc
att
act
atc
aca
ata
aac
aat 
ttt
taa
tcc
tac
tca
tct
tat
ttc
tta 


Permutations with Repetition of
Indistinguishable Objects:
Indistinguishable objects are
simply items (letters) that are repeated in the original set.
For example, if the word MOM was used instead of
CAT, in the example above, the two letter M's
are indistinguishable from one another, since they repeat.
Using MOM, some of our answers would have been duplicates of
one another because of the repeating M.
If we are looking for
answers that are not duplicates (unique answers), we must
deal with any letters (objects) that repeat in the original
set.
The number of
different permutations of
n
objects, where there are n_{1 }indistinguishable objects of
style 1, n_{2} indistinguishable objects of style 2, ..., and
n_{k }indistinguishable objects of
style
k, is ;

In general, repetitions are taken care
of by dividing the permutation by the factorial of the number of objects that are
identical.
Remember that
when n = r, _{n }P_{r} =
n!
So the formula above can also be seen as

