Solution: 2019 Winter Final - 4

Author: Michiel Smid

Question

Consider the sets $A = \{1,2,...,10\}$ and $B = \{1,2,...,14\}$. Let $S = \{(x,y) : x \in A, y \in B\}$. An element $(x,y)$ of $S$ is awesome, if $x$ is even or $y$ is even. What is the number of awesome elements in $S$?

(a)

105

(b)

107

(c)

104

(d)

106

Solution

Find A

Let A be the event that x is even

We choose 1 of the 5 even numbers in A: $\binom{5}{1}$

We choose 1 of the 14 numbers in B: $\binom{14}{1}$

$|A| = \binom{5}{1} \cdot \binom{14}{1} = 70$
Find B

Let B be the event that y is even

We choose 1 of the 10 numbers in A: $\binom{10}{1}$

We choose 1 of the 7 even numbers in B: $\binom{7}{1}$

$|B| = \binom{10}{1} \cdot \binom{7}{1} = 70$
Find $A \cap B$

Let $A \cap B$ be the event that x is even and y is even

We choose 1 of the 5 even numbers in A: $\binom{5}{1}$

We choose 1 of the 7 even numbers in B: $\binom{7}{1}$

$|A \cap B| = \binom{5}{1} \cdot \binom{7}{1} = 35$
Profit

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

$= 70 + 70 - 35$

$= 105$