Books that use probabilistic/combinatorial/graph theoretical/physical/geometrical methods to solve problems from other branches of mathematics I am searching for some books that describe useful, interesting, not-so-common, (possibly) intuitive and non-standard methods (see note *) for approaching problems and interpreting theorems and results in number theory, analysis, algebra, linear algebra, and other branches of mathematics. 
(*) Such methods can be (but not limited to) from the areas of 


*

*probability;

*combinatorics;

*graph theory;

*physics;

*geometry.


Examples of such books can be Uspenskii's Some Applications of Mechanics to Mathematics or Apostol's and Mnatsakanian's New Horizons in geometry.
 A: The Probabilistic Method by Alon and Spencer is a classic.
A: The classroom resource materials of MAA(mathematical association of america) can be very useful as it presents unusual approaches to mathematical ideas:
Some of its book I have used are as follows:
1.)Combinatorics 
A Problem Oriented Approach by  Daniel A. Marcus.
2.)Visual Group Theory by Nathan Carter.
3.)Exploratory Examples in Real Analysis by J.E.Snow and K.E.Weller.
You can find books suitable for you as you described in question from the link below: 
http://digital.ipcprintservices.com/publication/?i=140026&p=24
you'll enjoy maths through these books in above link!
A: Mark Levi, The Mathematical Mechanic: Using Physical Reasoning to Solve Problems. 
A: We're using   A Combinatorial Introduction to Topology by Henle as a supplemental resource in my general topology class right now.   Some of the alternate proofs are interesting,  like using Sperner's lemma to give a combinatorial proof of the Brouwer Fixed point theorem in $\mathbb R ^2$
