I am studying combinatorics, and at the moment I am having trouble with the logic behind more complicated counting problems. Given the following list of counting techniques, in which cases should they be used (ideally with a simple, related example):

* Repeated multiplication (such as $10 \times 9\times 8\times 7$, but not down to $1$)
* Addition
* Exponents
* Combination of the above ($2^6 + 2^5 + 2^4 + 2^3 + 2^2 + 2^1 + 2^0$)
* Factorials
* Permutations
* Combinations
* A case like this: $2^{10} \times \left({6 \choose 2} + {6 \choose 1} + {6 \choose 0}\right)$
* A case like this: $13 \times {4 \choose 3} \times {4 \choose 2} \times 12$
* A case like this: $13 \times {4 \choose 3} \times {4 \choose 2} \times {4 \choose 1}$

Sorry for the crash list of questions, but I am not clear on these issues, especially not good when I have a test in a few days!

Thank you for your time!