I've come across the following Combinatorics problems:
- A group of people go to an amusement park with $10$ rides, each of them goes on exactly $5$ rides and any two different people go on at most $2$ common rides. What is the highest possible number of people in the group?
- A grid with $3$ rows and $52$ columns is tiled with $78$ identical $2\times1$ blocks. In how many ways can this be done such that exactly two blocks are vertical?
Somehow Combinatorics has always been very difficult for me, I'm never sure what is the proper way to mathematically represent verbal problems like these in order to then solve them. I would really appreciate if someone could demonstrate and try to explain to me what is the general way of thinking involved in solving such problems.
In combinatorics there is a zoo of standard problems, some of them easy (e.g., "stars and bars"), some of them more difficult, but covered by some well known theory (e.g., "Polya theory"), and some of them really difficult, in particular when asymptotic results are desired (e.g., "partitions"). When confronted with a word problem you should first find out whether it belongs to, or can be reformulated such that it belongs to, a class of problems you are familiar with. If this is not the case and the involved numbers are small you can try to solve the problem by brute force, i.e., by systematically listing all cases. Maybe you can begin with the same problem in terms of smaller numbers, and if you are lucky you find a pattern.
The two problems presented in your question have nothing to do with each other.
The first is rather difficult and belongs to the field of "extremal combinatorics". Assume you want to prove that $n_{\rm max}=9$. This means that (a) you have to produce an admissible arrangement involving $9$ people, and (b) you have to prove that there are no admissible arrangements involving $10$ or more people. Here (b) is the difficult part. – I don't know how to tackle this problem in a systematic and generalizable way.
The second problem is rather easy, and you don't need big theories to solve it. An obvious handle are the two vertical pieces. In how many ways can you place them such that you can fill the rest of the rectangle with horizontal pieces? You'll find out that there is some specific "obstruction". A priori there are two cases to consider: Both vertical pieces are "blocking" the same two rows, or they are "blocking" two different pairs of rows.