Question

How many ways are there to arrange the letters of the word EDUCATION so that all the following three conditions hold?
- the vowels occur in the same order(EUAIO);
- the consonants occur in the same order(DCTN);
- no two consonants are next to each other.

• A. $15$
• B. $24$
• C. $72$
• D. $120$

Answer 1

The correct option is A $15$

$\_\_E\_\_U\_\_A\_\_I\_\_O\_\_$
We first fix the vowels in the desired order. This can be done in exactly one way.
Now, in the remaining six gaps, four consonants can be filled in ${}^6{C}_4$ ways. The desired order of consonants can be achieved in only one way.
$\therefore$ Required number of ways $=1\times {}^6{C}_4\times 1=15$