AMC10 2027 - Self-practice Set 1 Question 22

Question 22

Three distinct integers are selected at random from $1$ through $2016$, with all three-element sets equally likely. Let $p$ be the probability that their product is odd. Which statement is correct?

$p\lt \frac18$
$p=\frac18$
$\frac18\lt p\lt \frac13$
$p=\frac13$
$p\gt \frac13$