In how many ways can the letters of the word “INTERMEDIATE” be arranged so that:
Question:

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?

Solution:

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 $=\frac{6 !}{2 ! 3 !} \times \frac{6 !}{2 !}=21600$

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