2016 AMC 10A Problems/Problem 12

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?

$\textbf{(A)}\ p<\dfrac{1}{8}\qquad\textbf{(B)}\ p=\dfrac{1}{8}\qquad\textbf{(C)}\ \dfrac{1}{8}<p<\dfrac{1}{3}\qquad\textbf{(D)}\ p=\dfrac{1}{3}\qquad\textbf{(E)}\ p>\dfrac{1}{3}$

Solution

There are an equal amount of even and odd numbers from $1$ through $2016$ . This concludes that the probability of choosing an even number is the same of that of choosing an odd number. The only way to get a product of three integers to be odd is for all three integers to be odd. If the probability of choosing an odd number is $\frac{1}{2}$ , then the probability of choosing $3$ odd numbers would be $\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{8}.$ Therefore, the answer is $\boxed{\text{(B)}}$ .

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2016 AMC 10A Problems/Problem 12

Solution