Calculating the Number of Calculations in each Comparison

In your examples, the first type of "comparisons" are called $\textbf{combinations}$. They are selections of items characterized by distinct elements, such that the order does not matter. This is why, for example, once we have considered $(1,2)$, we have not to consider $(2,1)$. The formula is

$$C_k^n=\displaystyle \binom{n}{k}=\frac{n!}{k!(n-k)!}$$

where $n$ is the total number of items and $k$ is the size of each subset.

The second type of comparisons are called $\textbf{combinations with repetitions}$. They are selections of items characterized by not necessarily distinct elements, again such that the order does not matter. Compared with the combinations without repetitions, these combinations allow for duplicates and are often indicated using the multiset symbol. The symbol and the formula are

$$\left(\!\!\displaystyle \binom{n}{k}\!\!\right) =\binom{n+k-1}{k}=\frac{(n+k-1)!}{n!(k-1)!}$$

The third type of comparisons shown in the OP are called $\textbf{permutations with repetitions}$. They are ordering of not necessary distinct $k$ objects chosen in a set of $n$ items, such that the order does not matter. The formula is simply

$$n^k$$