Let us say that we are picking 2 letters from a set of 12. How would you describe the sample space?

Question

Let's say the first letter is from the following set: {A,B,C,D,E,F}. And the second letter is from the following set: {a,b,c,d,e,f}. Would this be simply $12\choose2$=66? Or would it be $_{12}P_2=132$? I tried to do it out and got $6$ ways to do the capital letter, 6 ways to do the lowercase letter, and then multiplied by 2 since $Aa\neq aA$. This was $6\times 6\times 2=72$ which is not equal to the answers I got using permutations and combinations. Which way is the correct method?

Answer 1

Let $S = \{A, B, C, D, E, F\}$; let $T = \{a, b, c, d, e, f\}$. Then the sample space is $$S \times T = \{(s, t) \mid s \in S, t \in T\}$$ that is, the set of ordered pairs in which the first element is a member of set $S$ and the second element is a member of set $T$. Since there are six choices for the first element and six choices for the second element, the number of such ordered pairs is $6 \cdot 6 = 36$.

Note that since the first element of the ordered pair is in set $S$ and the second element of the ordered pair is in set $T$, the outcome $aA$ is not a possible outcome.

The number $$\binom{12}{2}$$ represents the number of two-element subsets of a set with twelve elements.

You would get $72$ if you considered sequences of one uppercase and one lowercase letter if there were no requirement that the uppercase letter precede the lowercase letter.