The integer $134,678$ contains the $6$ distinct digits: $1, 3, 4, 6, 7, 8$.

Any arrangement of these $6$ digits forms a valid $6$-digit positive integer (since none of the digits is $0$, the leading digit is always nonzero).

The number of arrangements of $6$ distinct objects is:

$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$