Permutations
with Repetition
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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 n1 indistinguishable objects of style 1, n2 indistinguishable objects of style 2, ..., and nk 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 Pr n!
So the formula above can also be seen as


Example:

1.  How many different 5-letter words can be formed from the word   APPLE  ?

     =    54321      =   120    =    60 words
               21                   2

                                           You divide by  2!  because the letter  P  repeats twice.

                  

2.  How many different six-digit numerals can be written using all of the following six digits: 
      
4,4,5,5,5
,7

    =    65 4321     =    720   =   60
          21     321           12

                                       Two  fours repeat and  three  fives repeat, so divide by  2!  and  3!

 

 

See how to use your
TI-83+/TI-84+ graphing calculator  with permutations.
Click calculator.

                      


 
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Created by Lisa Schultzkie