How many words can be formed by arranging the letters of the word ‘ARRANGEMENT’, so that the vowels remain together?
To find: number of words where vowels are together
Vowels in the above word are: $A, A, E, E$
Consonants in the above word: R,R,N,G,M,N,T
Let us denote the all the vowels by a single letter say $Z$
$\Rightarrow$ The word now has the letters, R,R,N,G,M,N,T,Z
$\mathrm{R}$ and $\mathrm{N}$ are repeated twice
Number of permutations $=\frac{8 !}{2 ! 2 !}$
Now $Z$ is comprised of 4 letters which can be permuted amongst themselves
A and E are repeated twice
$\Rightarrow$ Number of permutations of $Z=\frac{4 !}{2 ! 2 !}$
$\Rightarrow$ Total number of permutations $=\frac{8 ! \times 4 !}{2 !^{4}}=60480$
The number of words that can be formed is 60480