CAT Permutation Combination: In how many ways can the letters of the word EDUCATION be rearranged such that the order in which the vowels appear in the word doesn’t change?
This is a classic question, so I am going to approach it in two ways.
Let us first find out how many ways the letters of the word EDUCATION can be rearranged. This is 9!.
Now, let us select some rearrangement of the word, say, DUECATONI. The word DUECATONI does not satisfy the condition that vowels should appear in the same order as in the word EDUCATION.
In this word, the vowels are in slots 2, 3, 5, 7, 9. A word with vowels in slots 2, 3, 5, 7 and 9 looks like this D _ _ C _ T _ N _ – the blanks are taken up by vowels. Now, the vowels can take up the slots in 5! ways. Within this 5! ways, only one order of vowels works for us – that is EUAIO. In other words, of the 5! words that are of the form D _ _ C _ T _ N _, only one D EU C A T IN O satisfies the second condition that the vowels should appear in one specific order. Or, of the total 9! rearrangements that are there for the word EDUCATION, only 9!/5! satisfy the second condition. So, the answer is 9!/5!
The word EDUCATION has 5 vowels and 4 consonants. Let us worry about these two separately. First, let us select the slots for the vowels. This can be done in 9C5 ways. In the remaining 4 slots, the 4 consonants need to go in – this can be done in 4! ways. The 5 vowels need to go into the 5 selected slots. This can be done in ONLY ONE way as the vowels’ order is already known. So, the number of ways is 9C5 * 4!.
9C5 * 4! is nothing but 9!/5! (otherwise, we would have been in trouble).