Textbook problem (quoted): In Texas Hold'em poker a player receives 2 cards first. These are called the player's pocket cards. Find the number of pocket card configurations (that is, the number of unordered samples of two cards from the deck of 52) that correspond to a pair. (The two cards have the same rank.)
My answer: $13 \cdot {4\choose 2} = 78$. First choose a rank out of the 13 possible ranks. Then choose two cards in no particular order from the four suits at that rank.
Question: What $n$-tuple am I applying the multiplication principle to in my answer? I mean I think I'm using the multiplication principle to count here, but that principle applies only to $n$-tuples. So am I framing all of the outcomes (the so-called sample space, I believe) as ordered pairs of the form
$$ (\text{rank}, \text{two-card pairs})? $$
I'm trying to make sure that I know exactly which concepts/theorems/facts from the text I am using/relying on as I am learning this stuff even if such formality seems overkill here. I want to be able to solve more complicated problems with justification moving forward.
Multiplication Principle (quoted): Suppose that a set of $n$-tuples $(a_1, \ldots , a_n)$ obeys these rules.
(i) There are $r_1$ choices for the first entry $a_1$.
(ii) Once the first $k$ entries $a_1, \ldots , a_k$ have been chosen, the number of alternatives for the next entry $a_{k+1}$ is $r_{k+1}$, regardless of the previous choices.
Then the total number of $n$-tuples is the product $r_1 \cdot r_2 \cdot r_3 \cdots r_n$.
Textbook: ''Introduction to Probability" by Anderson, Seppalainen, Valko. Appendix C on counting.