I need a systematic way to solve questions involving permutation and combinations or I need a way to verify my solution. Examples of problems: matching envelopes to people, how many people share a birthday, problems involving cards, etc. I know about probability axioms. The problem is calculating the probability of events from permutations and combinations. These two are concrete examples.
Eight indistinguishable rooks are randomly distributed on a chessboard. What is the probability that two rooks can attack each other? The correct answer is $\frac{8!}{64 \choose 8}$ but I got $\frac{2}{64 \choose 8}$ This happened because there I was not able to guess conditions where two rooks can attack each other despite not being on the same diagonal. This is under-counting.
Another example : There are 40 balls each with a number inscribed on them. Find the number of ways to choose 8 balls from the set such that only six of them have numbers between 1 to 8 inscribed on them. Correct answer is $\frac{{8 \choose 2} {32 \choose2}}{40 \choose 8}$ but i got $\frac{{8 \choose 2} {38 \choose2}}{40 \choose 8}$ because I did not remove balls I chose previously This was a case of over-counting.
I recognize these problems can be fixed by practice. I want to know if is there a way/algorithm to solve these problems with less involvement of logic using something like sets and or linear algebra.
If I apply the algorithm on the question involving rooks it should be immediately obvious that I am not considering the cases where two rooks can attack each other despite not being on a diagonal.
The twelvefold way is a good place to start. It provides a template for solving huge classes of elementary counting problems. Another useful, but more narrow, methodology is known as stars and bars.