The numbers 1 through 10 are recorded on individual slips of paper and placed in a hat. what is the probability of drawing the 3, not replacing it, and then drawing the 7?

First you have to determine the probability of drawing a 3 from the original 10 options.  You do this by take the 1 outcome you desire and divide it by all the options available:
1/10 = 0.1

Now you need to determine the probability of obtaining the 7 from the remaining 9 options.  You do this by taking the 1 outcome you desire from the and divide it by all the options available, which due to the fact that we've already taken 1 number out our remaining options are limited to 9 numbers:
1/9 = 0.11

Now add the 2 probabilities together to determine the probability of getting a 3 and then a 7.

0.1 + 0.11 = 0.21, which means the probability is 21%.