Question: 2018 Winter Midterm - 2

Author: Michiel Smid

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 $1,2,3,\dots,12$.
How many such strings have the property that

each even position contains an element of $\{a, b, c\}$, or
each odd position contains an element of $\{d,e\}$?

(a)

$3^{6} \cdot 5^{6} + 2^{6} \cdot 5^{6}$

(b)

$6^{3} \cdot 6^{5} + 6^{2} \cdot 6^{5}$

(c)

$6^{3} \cdot 6^{5} + 6^{2} \cdot 6^{5} - 6^{2} \cdot 6^{3}$

(d)

$3^{6} \cdot 5^{6} + 2^{5} \cdot 5^{6} - 2^{6} \cdot 3^{6}$