Answer
$151,200$ distinguishable permutations.
Work Step by Step
3 A's, 2 T's, 2 C's, 1 N, 1 R, 1 I. Total: 10 letters.
$\frac{10!}{3!\times2!\times2!\times1!\times1!\times1!}=\frac{10\times9\times8\times7\times6\times5\times4\times3\times2\times 1 }{6\times2\times2}=151,200$
