Question

Three distinct integers are selected at random between $1$ and $2016,$ inclusive. Which of the following is a correct statement about the probability $p$ that the product of the three integers is odd?

$p \lt \dfrac{1}{8}$

$p = \dfrac{1}{8}$

$\dfrac{1}{8} \lt p \lt \dfrac{1}{3}$

$p = \dfrac{1}{3}$

$p \gt \dfrac{1}{3}$

Solution:

The product is odd exactly when all three selected integers are odd. There are $1008$ odd and $1008$ even integers from $1$ to $2016$ .

Because the integers are selected without replacement, $p=\frac{1008}{2016}\cdot\frac{1007}{2015}\cdot\frac{1006}{2014}.$ The first factor is $\frac12$ , and each of the next two factors is slightly less than $\frac12$ . Therefore $p<\frac18$ .

Thus, the correct answer is A.

2016 AMC 10A Problem 12

Solution:

Problem 12 in Other Years