Question: 2017 Fall Midterm - 7
Let $n \geq 2$ be an integer. What does
$$
\sum_{k=2}^{n} {{n}\choose{k}} \cdot 2^{n-k}
$$
count?
(a)
The number of strings of length $n$, where each character is $a$ or $b$, that contain at least one $a$.
(b)
The number of strings of length $n$, where each character is $a$ or $b$, that contain at least 2 many $a$'s.
(c)
The number of strings of length $n$, where each character is $a$, $b$, or $c$, that contain at least one $a$.
(d)
The number of strings of length $n$, where each character is $a$, $b$, or $c$, that contain at least 2 many $a$'s.