<s>[INST] Die Roll Probability If Zachary rolls a fair die five times, what is the probability that the sum of his five rolls is 20?
1st I did:
Patterns of 5 that can give us 20
66611, 66521, 65531, 65522, 64442, 64433, 64415, 
Where do we go from here?
 [/INST] Assuming that you know generating functions.
We are interested in the coefficient of $x^{20}$ in the expansion $(x+x^2+x^3+x^4+x^5+x^6)^5$.
This is equivalent to the coefficient of $x^{15}$ in the expansion $\left( \frac{1- x^6}{1-x} \right)^5 $
The numerator is easily evaluated as
$$ 1- - 5 x^{6} + 10 x^{12} - 10x^{18} + 5 x^{24} - x^{30} $$
The denominator is 
$$ \sum x^i {i+4 \choose 4} $$
Hence, the answer is
$$ 10 \times {7 \choose 4} - 5 \times {13 \choose 4} + 1 \times {19 \choose 4} $$
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