How many solutions does the equation $x_1+x_2+x_3+x_4+x_5 = 8$ have?

Question

$x$_$1$+$x$_$2$+$x$_$3$+$x$_$4$+$x$_$5$ = 8

where $x$_$i$'s can take values $\{0,1,2,3\}$.

Answer 1

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 $\boxed{155}$.

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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.