Since you want those numbers to appear in that precise order, you only change is to extract 1 with the first draw, 2 with the second, and so on.
You have 10 slips at the beginning, so the probability of having 1 at the first draw is
[tex] \dfrac{1}{10} [/tex]
Now you have 9 slips left, so the probability of picking 2 with the second draw is
[tex] \dfrac{1}{9} [/tex]
Similarly, the remaining probabilities are [tex] \frac{1}{8} [/tex] and [tex] \frac{1}{7} [/tex]
You want these events to happen one after the other, and they are independent. So, the overall probability is the product of the single probabilities:
[tex] P = \dfrac{1}{10}\cdot\dfrac{1}{9}\cdot\dfrac{1}{8}\cdot\dfrac{1}{7} = \dfrac{1}{10\cdot9\cdot8\cdot7} = \dfrac{1}{5040} [/tex]
