If you know a formula that applies, use it. Otherwise start with a simpler problem, say draw 2 instead of 9, figure out how to solve it, then extend the method to 9.
Consider drawing 2. You draw 1 and it gets replaced in the deck so the odds of not getting the same card again are 51 in 52 or 51/52. Consider drawing 3. You draw 1 and the odds of not getting the same one on the 2d draw are still 51/52. Draw a 3d and the odds of getting neither the first card nor the second are 50 in 52 or 50/52. To say that both statements are true, that the second card is not the same as the first AND the third card is not the same as the first or the second, means to multiply the probabilities of each case alone, or 51/52 x 50/52. Keep going like this for 9 cards.