Could someone please explain counting to me? I'm taking a Discrete Math class and I understand the concepts behind counting, but I feel like I'm having to learn how to tackle word problems all over again.  I can't seem to figure out how to discern which tools I should be using.  Maybe I don't understand how the rule of sum, rule of product, permutations and combinations relate to one another, so it makes it difficult to figure out what I should be using for a particular scenario. Any help would be appreciated.  
 A: So this is a problem that can be tackled at a huge number of different levels. People write books on counting, with the purpose of explaining counting, and this discipline is called combinatorics. 
But to answer your question about sums and products and permutations and combinations. 
When you have the opportunity to make a single choice from one of two set, you sum the cardinalities of the two sets. This makes sense, because you're trying to find one from the collection of the two, seen as a single sets. For example, how many ways can you pick one person from a class of 4 boys and 3 girls? 
$7$. Because $3+4 = 7$. 
And for products, if you're making a sequence of independent choices, you can multiply them, because for each choice you make first, you can make one possibility from the second set of choices. So to pick one boy and one girl from our class of 4 and 3 boys and girls, we have 
$$4\cdot 3= 12$$
To find the ways of permuting something, we use the product rule we just described. A permutation is a rearrangement of an ordered tuple, so we need to number the ways of ordering, say $n$, things. We can set this up as subsequent choices. Our first choice is which of our $n$ goes in the first spot, the second is which of the $n-1$ go in the second, and so on. Because of this product rule, we can multiply this, and get the the number of permutations of an $n$ element set is 
$$n(n-1)\cdot\cdots(3)(2)(1) = n!$$
And for combinations, we use this when we're trying to find the number of subsets of a certain size of a, say $n$ element, set. There's an easy way of thinking of the formula for this. If you want to pick $k$ items from an $n$ element set, create 'boxes' like such
$$[\circ, \circ,  ... , \circ][\circ, \circ,  ... , \circ]$$
With $k$ spaces in the first and $n-k$ in the second, so there are $k$ total. Now we want to find a way to arrange the $n$ elements into these spaces, which by our previous discussion is $n!$. But wait! We count subsets in different orders as different subsets, which we can't allow. So we make it such that if all of the elements are the same in the first box, then two different permutations give the same combination. Well, for each way of putting $k$ in the first box, and $n-k$ in the second, there are $k!$ to arrange the first box, and $(n-k)!$ ways to arrange the second per our previous discussion. So we divide our way of distributing the $n$ elements into the boxes by the number of distributions each desired thing we want to count takes up from these, and get that our result is
$$\frac{n!}{k!(n-k)!}$$
And this encompasses the basics of combinatorics. But you can learn much more from books. If your class has a textbook, I recommend reading that. If not, there are a number of options. You might want to research your own, but I highly reccomend Graham/Knuth/Patashnik, Concrete mathematics. 
