Combinatorics: When To Use Different Counting Techniques 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!

Let me address some of the more general techniques on your list, since the specific ones just appear to be combinations of the general ones.
Repeated Multiplication: Also called "falling factorial", use this technique when you are choosing items from a list where order matters. For example, if you have ten flowers and you want to plant three of them in a row (where you count different orderings of the flowers), you can do this in $10 \cdot 9 \cdot 8$ ways.
Addition: Use this to combine the results of disjoint cases. For example, if you can have three different cakes or four different kinds of ice cream (but not both), then there you have $3 + 4$ choices of dessert.
Exponents: As with multiplication, but the number of choices does not decrease. For example, if you had ample supply of each of your ten kinds of flowers, you could plant $10 \cdot 10 \cdot 10$ different ways (because you can reuse the same kind of flower).
Factorials/Permutations: As with the first example, except you use all ten flowers rather than just three.
Combinations: Use this when you want to select a group of items from a larger group, but order does not matter. For example, if you have five different marbles and want to grab three of them to put in your pocket (so the order in which you choose them does not matter), this can be done in $\binom{5}{3}$ ways.
