I'm getting an answer choice (D), yet I might be wrong.
Instead of using casework, it's better to use complementary counting and caseowrk.
To start off, the number of ways to from seven-digit numbers such that no digits are repeating are: 9*9*8*7*6*5*4=544320 ways(numbers).
Next, imagine we have the following formation:
8 9 _ _ _ _ _
For the first blank, I have eight choices (excluding eight and nine, including zero)...then seven choices for the second blank, six for the third blank, and so on for 8*7*6*5*4=6720 numbers.
Note that the eight and nine are can be interchanged, so we have 2 * 6720=13440 ways(numbers).
Next, imagine eight and nine are in the second and third positions respectfully.
_ 8 9 _ _ _ _
There are seven choices for the first digit(excluding zero), seven for the fourth digit(including zero), six choices for the fourth digit and so on.
Thus, we have 7 * 7 * 6 * 5 * 4= 5880 cases.
This same case can happen in positions three and four, four and five, five and six, and finally six and seven.
5880 * 2 * 5=58800 ways(numbers).
Yes, thus, we have 544320-13440-58800=472080 ways(numbers) for choice (D).
Again, I might be wrong.
