How many words can be formed with the letters of the word ${}^{\prime }PATALIPUTR{A}^{\prime }$ without changing the relative order of the vowels and constants?

If we beak down the word $PATALIPUTRA$,
we have 6 consonants $\Rightarrow PTLPTR$
and 5 vowels$\Rightarrow AAIUA$
letter P takes ${}^6{C}_2$ places
T takes ${}^4{C}_2$ places out of remaining 4 places
L takes ${}^2{C}_1$ places out of remaining 2 places
R takes $1$ remaining place
Similarly,
Letter A takes ${}^5{C}_3$ places
I takes ${}^2{C}_1$ places out of remaining 2 place
U takes 1 remaining place
$\therefore$Total number of ways 
$=3600$ ways
Hence, option 'A' is correct.