Permutations with Repetition

Permutations
with Repetition
Topic Index | Algebra Index | Regents Exam Prep Center

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 n₁indistinguishable objects of style 1, n₂ indistinguishable objects of style 2, ..., and n_kindistinguishable 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, _nP_r = 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 ?

_{=    5·4·3·2·1
=   120    =
60 words


2·1
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

_{=    6·5 ·4·3·2·1
=    720   =   60


2·1 ·
3·2·1
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.