Question: 2018 Fall Final - 4
Consider strings of length 70, in which each character is one of the letters $a, b, c$. How many
such strings have at least 3 letters $c$?
(a)
$3^{70} - 2^{70} - 70 \cdot 2^{69}$
(b)
$3^{70} - 2^{70} - 70 \cdot 2^{69} - {70 \choose 2} \cdot 2^{68}$
(c)
$\sum_{k=4}^{70} {70 \choose k} \cdot 2^{k}$
(d)
$\sum_{k=4}^{70} {70 \choose k} \cdot 2^{70-k}$