In the book The Moscow Puzzles by Kordemsky I came across the following puzzle (no. 47):
Express 100 three ways with five 5s. You can use brackets, parentheses, and these signs: $+, -,\times, \div$.
Well, this is of course easy, after trying a bit you find:
$$100 = (5+5+5+5) \times 5 = 5 \times 5 \times 5 - 5 \times 5 = 5 \times 5 \times (5-5 \div 5)$$
Now, I wonder: Is there any theory behind this problem? Let me try to formulate an abstract problem:
Given $N \in \mathbb{N}$, a finite subset $S \subseteq \mathbb{N}$ and $n \in \mathbb{N}$ how many ways are there to express $N$ using $+, -,\times, \div$ and parentheses with $n$ numbers in $S$.
Of course we don't want to count obvious rearrangements or adding superfluous parentheses as new decompositions. It's probably easiest to start with $\# S = 1$. Even in the above explicit problem I cannot immediately say that there are exactly the three given solutions. I guess one could give some argument but all I can think of explicitly involves the given numbers $100$ and $5$, so it's nothing general.
Any ideas? There might also be feasible variants of my formulation above, it's just a first try.