The best way to learn combinatorics to do lots and lots of examples. When presented with an example in the text, read the problem and try and solve it by yourself before reading the solution. Also, the Supplement at the end of Chapter 5 contains detailed solutions of selected Exercises. Also, all odd problems have solutions in the back of the text.
Although working with other students is encouraged, you must be able to solve these problems yourselves. So it is best to work on them before discussing the solution.
Often one cannot ``check'' to see if an answer to a counting problem is correct. However, a very reliable method to check your work is to try and solve the problem a different way. If you get the same answer by a different approach, your chance of being correct is better. (See Example 4(e) on pages 183-184.)