How to find total number of permutations if repetition is allowed?
Answers
Given a set of objects such that there are identical objects of type 1, identical objects of type 2, , and identical objects of type , how many distinct permutations of the objects are there? Note that, in this case, all of the objects must appear in a permutation and two orderings are considered different if the two objects in some position are non-identical.
If the objects are all distinct, then we have seen that the number of permutations without repetition is . For each of these permutations, we can permute the identical objects of type 1 in possible ways; since these objects are considered identical, the arrangement is unchanged. Similarly, we can take any of the permutations of the identical objects of type 2 and obtain the same arrangement. Continuing this argument, we account for these repeated arrangements by dividing by the number of repetitions. This gives the following result for the total number of permutations: