The number of permutation of the string $MADEEASY$ in which all vowel are together are?
Here I think that $5!$ is the answer, but answer given $\frac{5!4!}{2!2!}$ Is changing of vowel position make a separate word? I mean when vowel are together, there is a word, but vowel together considered as a single letter, then why it's permutation makes a separate word?
Note all the vowels are $A,A,E,E$. Now removing these letters you transform the original word into- $MDSY$. Now, there are-$□M□D□S□Y□$ $=5$ gaps where you can add letters. Now group the letters $A,A,E,E$ together so that they always remain together, and treat them as one letter. So now you can arrange the leters in this group of vowels in $\frac{4!}{2!2!}$ ways(repeated permutations). And there were $5$ gaps to fill,also, the letters $M,D,S,Y$ can rearrange between themselves in $4!$ ways, so total number of ways $= \boxed{~~4! \cdot 5\cdot \frac{4!}{2!2!}~~} = \boxed{~~5!\cdot \frac{4!}{2!2!}~~}$