# The number of words in which the relative order of vowels and consonants remain unchanged.

If as many more words as possible be formed out of the letters of the word $DOGMATIC$ then compute the number of words in which the relative order of vowels and consonants remain unchanged.

Since the position of vowels and consonants is not to be changed,so the total ways are $3!\times 5!=720$

But the answer given is $719.$

• What is a word? In addition, presumably each letter must be used exactly once? I'm not entirely sure what is meant by 'the relative order of vowels and consonants remain unchanged' - does this mean the word must have $OAI$ and $DGMTC$ in that order? – Shuri2060 Jul 19 '17 at 9:42
• The way I interpret the problem statement, the answer would be ${8\choose 3}=56$, with valid words in alphabetical order being DGMOAITC, DGMOATIC, DGMOATCI, DGMOTAIC, DGMOTACI, ..., ODGMTCAI. However, your answer seems to count all words with consonant-vowel pattern CVCCVCVC, and the answer given seems to not count the original word. – Hagen von Eitzen Jul 19 '17 at 9:47
• Also, would it be possible to give the origin of this question? It is possible that $DOGMATIC$ is not counted, if your interpretation is correct (although to me, that would be an error by the question setter, given the wording). – Shuri2060 Jul 19 '17 at 9:51

Yes, the answer will be $719$ because you have to find the the number of new words formed since the question says,”as many more words as possible”. $720$ is the count including DOGMATIC, therefore new words formed will be $720-1=719$.