Number of words which can be formed with INSTITUTION such that vowels and consonants are alternate Question:

How many  words which can be formed with INSTITUTION such that vowels and consonants are alternate?

My Attempt:
There are total 11 letters in word INSTITUTION. The 6 consonants are {NSTTTN} and the 5 vowels are {IIUIO}.  So if we begin with consonant then we can have $6!$ different arrangement of consonants and $5!$ different arrangement for vowels. But I is repeated 3 times, T is repeated 3 times and N is repeated 2 times. Thus we get $$\frac{6! \cdot 5!}{3! \cdot 3! \cdot 2!} = 1200$$  different words.
Now, It is also possible that that word begins with a vowel, thus we will have another $1200$ words.
Thus total number of words formed is $1200 + 1200 = 2400$
But answer given is 1200. 
Am I missing something?  

If a word starts with a vowel, we will run out of vowel before we run out of consonant. 
Let C denote a consonant, and V a vowel, we have in the first case CVCVCVCVCVC, but in the second case we will have VCVCVCVCVC and we still have one C left. Which we can not add at the end. 
