For permutations with
repetition, order still matters.
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
allowing for repetition of the letters?
For this problem, 3 locations are
_____ • _____ •
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
__3___ • __3___ •
__3___ = 27 arrangements
Permutations with Repetition of
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
The number of
different permutations of
objects, where there are n1 indistinguishable objects of
style 1, n2 indistinguishable objects of style 2, ..., and
nk indistinguishable objects of
k, is ;
In general, repetitions are taken care
of by dividing the permutation by the factorial of the number of objects that are
when n = r, n Pr =
So the formula above can also be seen as