The right subforum for such questions is Discrete Math.

What about the operator that returns the truth value of the first proposition and ignores the second one? What about the negation of the second proposition? The negation of conjunction, disjunction?

This is easy to answer if you are familiar with truth tables. Since there are k arguments and one result, each row of the table has k + 1 truth values, the last of which is determined by the first k (given the operator, of course). How many rows are there? They must contain all possible combinations of k truth values. There are two possible values for the first argument; for each of those there are two possible values for the second argument, and so on.

Now, suppose there are n rows where n is a function of k. How many ways are there to fill the last column of the truth table, which determined the value of the operator on those k arguments? For each of the n slots, there are two options.