Question

Una rolls $6$ standard $6$ -sided dice simultaneously and calculates the product of the $6$ numbers obtained. What is the probability that the product is divisible by $4?$

$\dfrac34$

$\dfrac{57}{64}$

$\dfrac{59}{64}$

$\dfrac{187}{192}$

$\dfrac{63}{64}$

Solution:

Count the complement, where the product is not divisible by $4$ . This happens if the product has no factor of $2$ , or exactly one factor of $2$ .

All dice odd has probability $(\frac12)^6=\frac1{64}$ . Exactly one factor of $2$ means exactly one die is $2$ or $6$ , and the other five dice are odd. This has probability $6\cdot\frac26\cdot\left(\frac12\right)^5=\frac4{64}.$

The complement has probability $\frac5{64}$ , so the desired probability is $1-\frac5{64}=\frac{59}{64}$ .

Thus, the answer is C .

2021 AMC 10B Fall Problem 14

Solution:

Problem 14 in Other Years