Question: 2018 Winter Midterm - 3
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?
How many such strings contain at least two $a$'s?
(a)
$5^{12} - 4^{12} - 12 \cdot 4^{11}$
(b)
$5^{12} - 4^{12} - 12 \cdot 4^{12}$
(c)
$12^{5} - 12^{4} - 12 \cdot 11^{4}$
(d)
$12^{5} - 12^{4} - 12 \cdot 12^{4}$