This is an equation $c+o+n+e+t+i+u=8$, where $0\leq c\leq 3$, $0\leq o\leq 1$, $0\leq n\leq 2$, $0\leq e\leq 1$, $0\leq t\leq 2$, $0\leq i\leq 1$, $0\leq u\leq 1$.  We don't care about the order of the letters, so it looks like Stars & Bars, but with a lot of case work.

$c=0,1,2,3$, $n=0,1,2$, $t=0,1,2$, there are $36$ cases, but many of them are impossible, so it is not that bad.  If you find a better method, please share.