Daniel I A Cohen, Basic Techniques Of Combinatorial Theory, covers all the requested topics and more, and has a superb collection of exercises. To give you some idea, in the chapter on binomial coefficients, there are exercises leading you through a proof of Bertrand's Postulate and Chebyshev's estimates for the counting function for the primes.

I think that, unfortunately, the book is out of print.

Another fine book of the same vintage is Alan Tucker, Applied Combinatorics.