Question: 2018 Winter Midterm - 3

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 contain at least two $a$'s?

(a)

$5^{12} - 4^{12} - 12 \cdot 4^{11}$

(b)

$12^{5} - 12^{4} - 12 \cdot 12^{4}$

(c)

$5^{12} - 4^{12} - 12 \cdot 4^{12}$

(d)

$12^{5} - 12^{4} - 12 \cdot 11^{4}$