Solution: 2019 Fall Midterm - 5

Determine A

Let A be the event that a 4 letter string starts with $ad$

The first 2 letters are locked in place: 1 possibility

The last 2 letters can be a, b, c or d: $4^2$ possibilities

$|A| = 1 \cdot 4^2 = 16$

Determine B

Let be the event that a 4 letter string ends with $dcb$

The last 3 letters are locked in place: 1 possibility

The first letter can be a or d: $4^1$ possibilities

$|B| = 1 \cdot 4^1 = 4$

Determine $A \cap B$

Let A and B be the event that a 4 letter string starts with $ad$ and ends with $dcb$

The first 2 letters are locked in place: 1 possibility

The last 3 letters are locked in place: 1 possibility

$|A \cap B| = 1 \cdot 1 = 1$

Profit

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

$A \cup B = 16 + 4 - 1 $

$A \cup B = 19$