How many solutions does the equation $x_1+x_2+x_3+x_4+x_5 = 8$ have? $x$$1$+$x$$2$+$x$$3$+$x$$4$+$x$$5$ = 8
where $x$$i$'s can take values $\{0,1,2,3\}$.

In this case, it's easy to check by hand that the only solutions are 
\begin{align*}
   0+0+2+3+3&=8\\
   0+1+1+3+3&=8\\
   0+1+2+2+3&=8\\
   0+2+2+2+2&=8\\
   1+1+1+2+3&=8\\
   1+1+2+2+2&=8,
\end{align*}
up to permutations of the terms in the different sums. Thus the number of solutions is 
$$\frac{5!}{2!\,2!}+\frac{5!}{2!\,2!}+\frac{5!}{2!}+\frac{5!}{4!}+\frac{5!}{3!}+\frac{5!}{2!\,3!} = \boxed{155}.$$
It's easier to order the set of quintuples (say lexicographically like I did) when coming up with these.
Alternatively, you can construct a generating function for the problem, in this case $(1+x+x^2+x^3)^5$ is suitable, and study the coefficient of $x^8$, which is $\boxed{155}$. 

If you are not familiar with generating functions, read this short example of how they can be used to solve counting problems. They are a very useful tool when it comes to solving counting problems like this one.
