I am searching for book on discrete mathematics which is suitable for self study. This mean I want it to have exercises with answers (It would be ideal if it had solutions). I have already read "Discrete Mathematics" of Kenneth Ross. I have also partially read "Concrete Mathematics" of Knuth but I didn't like the style much. I am searching for next book to read. I do not have any requirements on topics I just want it to cover few of them like combinatorics and counting, recurrences and probably generating functions.
Epp's text on Discrete Mathematics is a very nice read. Johnsonbaugh is good as well, but is more technical and more geared towards computer scientists.
Nicholas Loehr's text Bijective Combinatorics is a great read for the topics you listed, which fall in the realm of combinatorics. Loehr's text is rigorous and thorough, but it is also very well written and intuitive. I did my undergraduate work at Virginia Tech where he teaches, and he has a reputation for being both brilliant and a fantastic instructor. His book lives up to that reputation, in my opinion.
Make sure you avoid Alan Tucker's Applied Combinatorics textbook. It's a horribly written book, and the only redeeming quality are the exercises.
I'm taking a Discrete Mathematics course right now, using Applied combinatorics by Roberts and Tesman. It does have some good 'real life' examples and applications.
Here are very complete notes and problem sets from Jacob Lurie's (Harvard) combinatorics course (Harvard).