Answer
$9,459,450$ DNA sequences.
Work Step by Step
Total: $2+4+4+5=15$ letters.
It is a Permutation of 15 letters, (two of a kind, four of a second kind, four of a third kind and five of a fourth kind):
$\frac{15!}{2!\times4!\times4!\times5!}=\frac{15\times14\times13\times12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1}{2\times24\times24\times120}=9,459,450$
