Question

"director" how many arrangements are possible where the vowels will not change their order?

to solve this problem i have been taught to assume the 3 vowels same letter and then to do permutations. so it is 8!/(3!*2!).

but my question is if i solve the problem like this
A) i will assume the three vowels 1 letter, then total number of letter is 6. and the permutation is 6!/2!. what's my mistake here.
if the question says that the vowels will stay together then i WOULD REPEAT THE A) MEANS PREVIOUS STEP AND THEN DO THE INDIVIDUAL ARRANGEMENT OF THE VOWELS WHICH IS 3! AND THE TOTAL NUMBER WOULD BE 3!*(6!/2!).

i have skipped just the last step to solve my main problem. so what's my mistake.please explain.

Answer 1

A multiset permutation is an ordered arrangement of elements of M in which each element appears exactly as often as is its multiplicity in M. An anagram of a word having some repeated letters is an example of a multiset permutation.In our case we should use a formula of multiset permutation, so:P = 6!/(2!*3!) = 60(3! as the vowels will not change their order).