How to understand the rule of product in discrete mathematics?

Question

Proposition 2.6 (Rule of Product). Let T be a set of ordered k-tuples ($a_1$, ..., $a_k$), with the property that there are $r_i$ choices for each coordinate between 1 $\leq$ i $\leq$ k. Then |T| = $r_1$$r_2$ ... $r_k$.

I am taking an introductory discrete mathematics course, and we are learning the cardinality of sets in the form of the product.

This is one of the propositions I read in my lecture notes, I am having a hard time understanding its content.

Suppose I have a set of an ordered 2-tuples called S, if I understand this correctly, S is like S = {(1, 1), (1, 2), (2, 1,), (2, 2)}, so in this case, |S| = 4. I don't understand the part in the propostion that says we have $r_i$ choices for each coordinate between 1 $\leq$ i $\leq$ k, what does this coordinate mean? Is it each tuple in my set S? In addition, what does "we have $r_i$ choices for each coordinate between 1 $\leq$ i $\leq$ k" mean? Why is it $r_i$ choices?

Thanks.

Answer 1

They are not coordinates in the sense you learn in calculus.

Suppose you have to count the pairs $(A,B)$ where $A$ is a color (red, white or blue) and $B$ is a shape, say circle or start. In the rectangle

       circle   star
red       *       *
white     *       *
blue      *       *

you can think of "shape" as a coordinate on the horizontal axis and "color" as one on the vertical axis. With that picture it's easy to see $2 \times 3 = 6$.