Solution: 2015 Winter Midterm - 5

Author: Michiel Smid

Question

Consider strings of length 99 consisting of the characters $a$, $b$, and $c$. How many such strings are there that start with $abc$ or end with $bbb$?

(a)

None of the above.

(b)

$3^{99} - 2 \cdot 3^{96}$

(c)

$2 \cdot 3^{96} - 3^{93}$

(d)

$3^{96} + 3^{96}$

Solution

Let A be the set of strings that start with $abc$.

|A| = first 3 letters are set in stone while other 96 letters can be anything.

Thus, there are $3 \cdot 3 \cdot 3 \cdot 3^{96}$ strings in A.

Let B be the set of strings that end with $bbb$.

|B| = last 3 letters are set in stone while other 96 letters can be anything.

Thus, there are $3^{96} \cdot 3 \cdot 3 \cdot 3$ strings in B.

Let $A \cap B$ be the set of strings that start with $abc$ and end with $bbb$.

$|A \cap B|$ = first 3 letters and last 3 letters are set in stone while other 93 etters can be anything.

$|A \cup B| = |A| + |B| - |A \cap B|$

$|A \cup B| = 3^{96} + 3^{96} - 3^{93}$

$|A \cup B| = 2 \cdot 3^{96} - 3^{93}$