In how many ways can the letters of the word "INTERMEDIATE" be arranged so that:
(i) the vowels always occupy even places?
(ii) the relative order of vowels and consonants do not alter?
The word INTERMEDIATE consists of 12 letters that include two Is, two Ts and three Es.
(i) There are 6 vowels $(I, I, E, E, E$ and $A)$ that are to be arranged in six even places $=\frac{6 !}{2 ! 3 !}=60$
The remaining 6 consonants can be arranged amongst themselves in $\frac{6 !}{2 !}$ ways, which is equal to 360 .
By fundamental principle of counting, the number of words that can be formed $=60 \times 360=21600$
(ii) The relative positions of all the vowels and consonants is fixed.
Arranging the six vowels at their places, without disturbing their respective places, we can arrange the six vowels in $\frac{6 !}{2 ! 3 !}$ ways.
Similarly, arranging the remaining 6 consonants at their places, without disturbing their respective places, we can arrange the 6 consonants in $\frac{6 !}{2 !}$ ways.
By fundamental principle of counting, the number of words that can be formed $=\frac{6 !}{2 ! 3 !} \times \frac{6 !}{2 !}=21600$