8. [5] A single burger is not enough to satisfy a guy's hunger. The five guys go to Five Guys' Restaurant, which has 20 different meals on the menu. Each meal costs a different integer dollar amount between $\$ 1$ and $\$ 20$. The five guys have $\$ 20$ to split between them, and they want to use all the money to order five different meals. How many sets of five meals can the guys choose?
Answer: 7 Suppose the meals, sorted in descending order, cost $5+x_{1}, 4+x_{2}, \ldots, 1+x_{5}$. To satisfy the conditions in the problem, the $x_{i}$ must be a non-increasing sequence of non-negative integers which sums to 5 . Therefore, there is exactly one order for each partition of 5: order the elements of the partition from largest to smallest and use these parts as the $x_{i}$. For example, the partition $3+2$ corresponds to the order $5+3,4+2,3,2,1$. There are thus 7 orders, corresponding to the 7 partitions of 5 below.
$$
1+1+1+1+1,1+1+1+2,1+2+2,1+1+3,2+3,1+4,5
$$

These partitions yield the following seven orders:
$$
\begin{array}{c}
(2,3,4,5,6),(1,3,4,5,7),(1,2,4,6,7),(1,2,3,5,7) \\
(1,2,3,6,8),(1,2,3,5,9),(1,2,3,4,10)
\end{array}
$$