2021 Fall AMC 12B Problems/Problem 11
- The following problem is from both the 2021 Fall AMC 10B #14 and 2021 Fall AMC 12B #11, so both problems redirect to this page.
Problem 11
Una rolls 
standard -sided dice simultaneously and calculates the product of the 
numbers obtained. What is the probability that the product is divisible by 
Solution
We will use complementary counting to find the probability that the product is not divisible by 
. Then, we can find the probability that we want by subtracting this from 1. We split this into 
cases.
Case 1: The product is not divisible by .
We need every number to be odd, and since the chance we roll an odd number is 
our probability is 
Case 2: The product is divisible by , but not by .
We need 
numbers to be odd, and one to be divisible by , but not by . There is a 
chance that an odd number is rolled, a 
chance that we roll a number satisfying the second condition (only and work), and ways to choose the order in which the even number appears.
Our probability is 
Therefore, the probability the product is not divisible by is 
.
Our answer is 
.
~kingofpineapplz ~minor edits by stjwyl
See Also
| 2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 13 |
Followed by Problem 15 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
| 2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 10 |
Followed by Problem 12 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
