Answer
$138,600$ DNA sequences.
Work Step by Step
Total: $1+4+3+4=12$ letters.
It is a Permutations with Nondistinct Letters:
$\frac{12!}{1!\times4!\times3!\times4!}=\frac{12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1}{1\times24\times6\times24}=\frac{479,001,600}{3456}=138,600$
