My question: Can you help me understand counting rules? I do not understand the application of different counting rules for use in probability. I have a problem that I THINK I solved by counting by hand, but I want to use the formulas given by my teacher, listed at the bottom of my question. I especially don't understand formulas 4.) and 5.) but I think I need to use one of them.
An example: A secretary types three letters and the three corresponding envelopes. In a hurry, he places at random one letter in each envelope. What is the probability that at least one letter is in the correct envelope?
So, am I thinking permutations here? Combinations? One of the other three rules or some combination? What's the difference between "distinct", "unique"? How do these words apply to this context? How does the term "order" apply to this context. There are seem to be elements of two "kinds" here, and maybe "partitions"? Does order matter? I am having a very hard time understanding what these how these vocabulary words apply to this seemingly simple problem.
My work: I count 9 possible ordered pairs (2-tuples?). I think that order does not matter, i.e. (letter,envelope) = (envelope,letter). I also note that each element of the ordered pair must be of a unique kind. I remember that the probability of at least one correct ordered pair is $P(C)=1-(C_N)$, where $C_N$ is the event that no correct combinations occur.
Now I count 6 combinations that are incorrect, out of 9 total. So again, just counting I think that the probability of the event at least one correct combo is $P(C)= 1 -{6 \choose 3}/{9 \choose 3}$, but this does not yield an answer that matches what I find online.
The formulas: I have been given 5 counting rules to assist me in my probability theory homework, 3 of which have titles, and then two more for which the teacher did not have a title or name, only an equation and a confusing (for me) description of use. I am consistently at a loss to match the type of problem to the correct formula(s). I cannot find the last two rules in my textbook or by googling.
1.) Permutations
2.) Combinations
3.) Multiplication rule
4.) Let $r_1,r_2,\dots,r_k \in \mathbb{Z}^+: \sum_{i=1}^{k}r_i=n$, then the number of ways n elements can be partitioned into k groups in which group $i$ contains $r_i$ is:
$$n!/(r_1!r_2!\dots r_k!)$$
5.) The number of distinct permutations of length k from a set of n objects where each object in the permutation is unique in kind is: $$n!/(n_1!n_2!\dots n_k!)$$