What exactly is correct way (mathematically) to express a combination?
For instance if I have $3$ objects $O_1$, $O_2$, $O_3$ so if I simply write $O_1O_2O_3$, it's impossible to tell whether what I have here is a permutation or a combination.
I understand one way to counter this problem would be to simply mention whether it's a combination or a permutation but I was wondering, is there a mathematical notation as such to write a permutation or a combination without ambiguity?
I thought of writing writing combination as a set like $\{ O_1,O_2,O_3 \}$ which conveys the point that order isn't necessary. Similarly I could use tuples for permutations but my professor told me it's a bad idea, he did explain it briefly (something along the lines a combination isn't exactly a set...blah blah) but I honestly didn't understand him, hence this question.
Hoping someone would give me a better way to visualise and express combinations and permutations (I understand what they are, just can't properly write it out because of notation problems)
Thank you!
You are right and your professor is wrong - the combination $n \choose k$ is exactly the number of $k$-element subsets of a set with $n$ elements. In many combinatorics textbooks, that is the actual definition given.
Your notion of tuples as permutations isn't going to work. The issue is that $(1, 1, 1)$ is a perfectly valid triple of $\{1,2,3,4,5\}$, but it is not a permutation because of the repeated values. An accurate way of thinking about is that the permutation $_3P_5$ is the number of injective (i.e. one-to-one) mappings from $\{1,2,3\}$ to $\{1,2,3,4,5\}$. Then the factorial $5!={_5P_5}$ is the number of bijections on $\{1,2,3,4,5\}$.