Question: 2017 Fall Final - 8
Consider strings consisting of characters, where each character is one of the letters $a$, $b$, and
$c$.
For any integer $n \geq 1$, let $E_n$ be the number of such strings of length $n$ that contain an even number of $c$'s, and let $O_n$ be the number of such strings of length $n$ that contain an odd number of $c$'s. (Remember that 0 is an even number.)
Which of the following is true for any integer $n \geq 2$?
For any integer $n \geq 1$, let $E_n$ be the number of such strings of length $n$ that contain an even number of $c$'s, and let $O_n$ be the number of such strings of length $n$ that contain an odd number of $c$'s. (Remember that 0 is an even number.)
Which of the following is true for any integer $n \geq 2$?
(a)
$E_n = 2 \cdot O_{n-1} + E_{n-1}$
(b)
$E_n = 2 \cdot O_{n-1} + O_{n-1}$
(c)
$E_n = 2 \cdot E_{n-1} + E_{n-1}$
(d)
$E_n = 2 \cdot E_{n-1} + O_{n-1}$