Question 767679
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X a X e X i X o X u X

In the above there are 6 positions marked with X's in which 
we can insert consonants q, t, n.  

We can choose 3 of those 6 positions to insert one consonant
each in C(6,3) ways.          Example:   a <font color="red">t</font> e i <font color="red">n</font> o u <font color="red">q</font>

We can choose 2 of those 6 positions to insert two consonants 
in the leftmost one and one consonant in the rightmost one
in C(6,2) ways                Example:   a e i <font color="red">q n</font> o <font color="red">t</font> u 

We can choose 2 of those 6 positions to insert one consonant 
in the leftmost one and two consonants in the rightmost one
in C(6,2) ways                Example:   <font color="red">n</font> a e i <font color="red">t q</font> o u

We can choose 1 of those 6 positions to insert all three
consonants in C(6,1) ways     Example:   a e i <font color="red">n q t</font> o u 

So that's C(6,3)+C(6,2)+C(6,2)+C(6,1) = 20+15+15+6 = 56 ways
to choose places to insert consonants.

For each of those 56 ways to insert 3 consonants, the consonants
can be arranged in 3! or 6 ways.

Answer 56·6 = 336 ways.

Edwin</pre>