Agreed!

What is the number of permutations when all objects are not distinct and repetition is allowed when taken all or some at a time.

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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 ` ` then there would be 6 arrangements as ` ` 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:

` `  Here ` ` is the number of repetitions of the element ` ` . For mom we have ` ` . (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 ` ` . (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!.)

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The formula for the number of arrangements with repetitions is :

` `

`(n!)/(n_1!n_2!***n_k!)`

` `

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If you take just some of the objects, the same thinking applies.

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The formula for the number of arrangements of n objects taken r at a time, allowing for repetitions, is:

`(_nP_r)/(n_1!*n_2!***n_r!)`

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