Explain and give the formula for permutations of objects when all objects are not distinct and repetition is allowed

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embizze | High School Teacher | (Level 2) Educator Emeritus

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The number of permutations is the number of ways the elements of a set can be listed where order matters. Thus the number of permutations of the letters cat is 6: it is easy enough to list the possibilities: act.atc,cat,cta,tac,tca.

However, if the original contains indistinguishable elements, such as the set mom, you are no longer dealing with permutations. However, you can determine the number of possible sets:
With such a small set, you can list the possible arrangements:omm,mom,mmo. Notice that there are 3, not 6, "permutations" (arrangements) of the three letters. If the letters had been given as `m_1m_2o` then there would be 6 arrangements as `m_1",and,"m_2` are distinguishable. We divided by the number of arrangements of the letters m, or divided by 2.

The general formula for the number of arrangements of n letters containing repeats is:

`(n!)/(n_1!*n_2!***n_k!)`  Here `n_1` is the number of repetitions of the element `n_1` . For mom we have `(3!)/(2!1!)=6/2=3` . (For completeness I included the number of repetitions of each object -- here there is 1 "o", and 1!=1 by definition.)

For mississippi we have `(11!)/(1!*4!*4!*2!)=39916800/(24*24*2)=34650` . (Note that there are 4 "s's"; there are 4! ways the esses could have been placed; as they are indistinguishable you divide by 4!.)


The formula for the number of arrangements with repetitions is :