Thinking Mathematically (6th Edition)

$105$ ways
See Permutations of Duplicate Items, page 699. The number of permutations of $n$ items, where $p$ items are identical, $q$ items are identical, $r$ items are identical, and so on, is $\displaystyle \frac{n!}{p!q!r!\ldots}$ ---------------- 5,446,666 contains n=7 digits, of which there are 4 duplicate 6s ... p=4, 2 duplicate 4s ... q=2 a single 5 ..... in all n=7 (checking) ... So, the total number of distinct permutations is $\displaystyle \frac{7!}{4!2!}=\frac{7\times[6]\times 5\times(4\times 3\times 2\times 1)}{(4\times 3\times 2\times 1)\times[2]}$ $=7\times 3\times 5$ $=105$ ways