Question:
Find the number of arrangements of the letters of the word ‘ALGEBRA’ without altering the relative positions of the vowels and the consonants.
Solution:
To find: number of arrangements without changing the relative position
The following table shows where the vowels and consonants can be placed
Consonants can be placed in the blank places
There are 3 spaces for vowels
There are 3 vowels out of which 2 are alike
Vowels can be placed in $\frac{3 !}{2 !}=3$ ways
There are 4 consonants, and they can be placed in $4 !=24$ ways
$\Rightarrow$ Total number of arrangements $=24 \times 3=72$ ways
72 arrangements can be made