2018 AMC 10 A
Complete problem set with solutions and individual problem pages
When fair standard sided die are thrown, the probability that the sum of the numbers on the top faces is can be written as , where is a positive integer. What is ? (2018 AMC 10A Problem, Question#11)
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The minimum number that can be shown on the face of a die is , so the least possible sum of the top faces of the dies is .
In order for the sum to be exactly , to dices' number on the top face must be increased by a total of .
There are ways to do so: , , and .
There are for Case , for Case , and for Case .
Therefore, the answer is .
Rolling a sum of with dice can be represented with stars and bars, with stars and bars. Each star represents one of the dots on the die's faces and the bars represent separation between different dice. However, we must note that each die must have at least one dot on a face, so there must already be stars predetermined. We are left with stars and bars, which we can rearrange in ways.
Add possibilities. There are ways to sum to , listed below.
,,,,,,; ,,,,,; ,,,,,.
Add up the possibilities: .
Thus we have repeated Solution exactly, but with less explanation.
We can use generating functions, where is the function for each die. We want to find the coefficient of in , which is the coefficient of in . This evaluates to .
If we have every number at its minimum, it would be as a sum. So we can do to find the amount of balls we need to distribute to get three more added to the minimum to get , so the problem is asking how many ways can you put Balls into boxes. From there it is .
