How many words can be formed by arranging the letters of the word
Question:

How many words can be formed by arranging the letters of the word ‘ARRANGEMENT’, so that the vowels remain together?

Solution:

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