#### Answer

$105$ ways

#### Work Step by Step

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
$\displaystyle \frac{n!}{p!q!r!\ldots}$
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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