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psilentist
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Homework Statement
If you roll a fair die 5 times, what is the probability that the sum of the five rolls is 20?
Homework Equations
The Attempt at a Solution
I know the sample space is all possible permutations of 1 to 6 taken five at a time with repetition, which is 6^5 = 7776.
And I know this can be done with the principle of inclusion and exclusion.
The problem can be seen as distributing 20 identical objects into 5 labeled containers such that each has between 1 and 6 (inclusive) objects.
This reduces to distributing 15 objects such that each container has 0 to 5 (inclusive) objects.
So using PIE, the base set is all solutions to 15 objects into five labeled containers, C(15 + 5 - 1, 15), and the condition is that a container has 6 or more objects.
There are C(5, 1) = 5 ways the condition can be satisfied once, and in each case there are C(10 + 5 - 1, 10) solutions.
There are C(5, 2) = 10 ways the condition can be satisfied twice, and in each case there are C(5 + 5 - 1, 5) solutions.
So the answer should be [C(15 + 5 - 1, 15) - 5*C(10 + 5 - 1, 10) + 10*C(5 + 5 - 1, 5)]/7776 = 131/7776.
But the book says 651/7776.
Can anyone see a problem with what I'm doing?