Binomial theorem. I first saw this thing (admittedly much to late in life) in a third year class entitled non-linear dynamics and chaos theory.  There if i am remembering correctly we used to look for non-zero terms to figure out what kind of sink/source we had basically keep going through the theorem until you found a non-zero term and you could use that to figure out what happened near a sink/source. 
I have seen a few formulas of this theorem in my time one obviously being the single variable and one being the multivariable theorem the last one i have seen im less sure what to call it and i have never used it.
$ \dfrac {n!}{k!(n-k)!} $ which is equivalent to $ \dfrac {n(n-1)\cdots(n-k+1)} {k!}$ 
Which honestly took me awhile to figure out that these two were equivalent. 
Noticing $ 0 \leq k \leq n $ where it was written and realizing $n$ and $k$ were integers helped a lot.
I guess what im looking for is some material/book with some problems (and preferably worked out solutions) involving the binomial theorem being applied in different ways and forms any recommendations appreciated.
 A: I did a quick search and came across this thread on combinatorics textbooks. Alan Tucker's Applied Combinatorics text is suggested a lot. I used that one in my 3000 class on combinatorics and graph theory. It is good for providing you lots of problems, but not much else. I wouldn't use that as my primary source. Nick Loehr's text on Bijective Combinatorics is a stronger book and better written, but it is also geared towards advanced undergraduates and beginner graduate students. The first few chapters might be good to investigate.
A standard Discrete Math text would also have some introductory material on counting. Rosen, Epp, and Johnsonbaugh are good textbooks to look into here. 
Note that the binomial coefficient counts the number of ways to form a k-element subset from $n$ elements. It counts combinations.
A: The Wikipedia page on binomial coefficients is quite useful:
http://en.wikipedia.org/wiki/Binomial_coefficient
So is the mathworld page on the same:
http://mathworld.wolfram.com/BinomialCoefficient.html
You might find Graham, Knuth and Patashnik's Concrete Mathematics quite interesting as well.
