Please explain: How many such strings have the following properties

Question

We consider strings consisting of 12 characters, where each character is an element of the set {a, b, c, d, e}. The positions in such strings are numbered as 1,2,3,...,12.

How many such strings have the property that each even position contains an element of {a, b, c}, and each odd position contains an element of {d, e}?

I know that it is 6^6, but how do we get that?

I initially thought it was 6^2 x 6^3 because every even pos has a total of 6^3 choices and every odd pos has 6^2 choices, but I guess that is wrong.

Answer 1

Consider options:

$$\{d/e\}\{a/b/c\}\{d/e\}\{a/b/c\}\{d/e\}\{a/b/c\}\{d/e\}\{a/b/c\}\{d/e\}\{a/b/c\}\{d/e\}\{a/b/c\}$$

Counting options: $$2\cdot 3\cdot2\cdot 3\cdot2\cdot 3\cdot 2\cdot 3\cdot2\cdot 3\cdot2\cdot 3=6^6$$

Answer 2

You were quite close. You just had your exponentials backwards. It's $3^6$ total options for the even positions ($3$ for each even position), and $2^6$ for each odd position, so the total is $3^6 \cdot 2^6 = 6^6$.