Letters Arrangement

[mathsishardd]

We have the word: RAPSEVNOBLURKANIWA

We want to find all unique permutations that could be made with RAP always staying together, so that there are no 3 adjacent consonants, no 2 adjacent vowels and for every odd pair of adjacent consonants, the first letter is alphabetically smaller than the second, while for every even pair - the opposite (the first letter is alphabetically bigger than the second).

No single idea how to solve this, computer program would take atleast few months to solve.

 

[pka]

We have the word: RAPSEVNOBLURKANIWA
We want to find all unique permutations that could be made with RAP always staying together, so that there are no 3 adjacent consonants, no 2 adjacent vowels and for every odd pair of adjacent consonants, the first letter is alphabetically smaller than the second, while for every even pair - the opposite (the first letter is alphabetically bigger than the second).

This is just a true busy-work if there were ever one. Moreover, I think its author may have written some internal contradictions into it.
First of all it is not clear if the triple \(\displaystyle RAP\) must stay in that order or can be permuted,, so lets assume that is a fixed three-sum.
Out of the eighteen letters in the given string there are two A's & two N's (there is no need to count the repeating letters in RAP).
\(\displaystyle \_\_\_\_\boxed{RAP}\_\_\_\_S\_\_\_\_V\_\_\_\_N\_\_\_\_L\_\_\_\_B\_\_\_\_R\_\_\_\_K\_\_\_\_N\_\_\_\_W\_\_\_\_\)
Now using the ten non-vowels to create eleven blank spaces into which we may put one vowel to separate them from one another.
The separators can be arranged in \(\displaystyle \frac{10!}{2!}\) ways.(Because there are two N's we must divide by 2!)
There are \(\displaystyle \dbinom{11}{6}\) places to put the six vowels and \(\displaystyle \frac{6!}{2!}\)ways to arrange them.

I really don't understand what "odd pair of adjacent consonants" could possibly mean.
I hope that I have given you a start.

 

[mathsishardd]

After every consonant there could be vowel or consonant ( not 3 consonants in a row tho ), for the odd/even pairs - examples: in the given word: (1-odd)-P is smaller than S, (2-even)-V is bigger than N, (3-odd)-B is smaller than L, (4-odd)-R is bigger than K