Compound Probability

Alicia writes the number 1 to 45 on separate cards. She then randomly
chooses three of the cards. What is the probability that the 2nd and
3rd cards will include the digit 9 in the number.

Using a permutation solution method, I find the total number of
possible outcomes (45_P_3, 45 cards taken 3 at a time) which gives
85140 different arrangements of the 3 numbers.

I also use permutations to find the total number of ways the 4
numbers (9, 19, 29, and 39) can be arranged taken two at a time
(4_P_2) to get 12 different arrangements. (These are the
arrangements of the LAST 2 numbers with a 9). For each of the 12
arrangements, there are 43 different numbers that can come first
in the 3 letter group, so 12 * 43 = 516 total arrangements
(desired outcomes)

Using the basic probability model I divide the number of desired
outcomes by the number of possible outcomes.

516/85140 = .00606 = .606%

*I have even made lists to check that this is correct.*

However, the answer in the book uses the multiplication rule
(which seems valid) to find the solution.

P(any card the fist time) * P(9 the 2nd time) * P(9 the 3rd time)

=45/45 * 4/44 * 3/43

= .00634 = .634%

This doesn't agree with the permutation method (which I trust more
unless you can find a flaw).

However, I can get the same solution by multiplication if I start
with 45 in the denominator.

4/45 * 3/44 = .00606 = .606%

But I can't find any logic at all in why you would do this.
Can you shed any light on this for me?

THANKS!