#### Answer

$630$ 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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5,432,435 contains n=7 digits, of which there are
2 duplicate 5s ... p=2,
2 duplicate 4s ... q=2
2 duplicate 3s ... r=2
a single 2
..... in all n=7 (checking) ...
So, the total number of distinct permutations is
$\displaystyle \frac{7!}{2!2!2!}=\frac{7\times 6\times 5\times(4)\times 3\times[2]\times 1}{(2\times 2)\times[2]}$
$=7\times 6\times 5\times 3$
$=630$ ways