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?

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
$$ 10 \times {7 \choose 4} - 5 \times {13 \choose 4} + 1 \times {19 \choose 4} $$
