  Let $f_1,...,f_k:M\to N$ be maps between closed manifolds, $N(f_1,...,f_k)$
and $R(f_1,...,f_k)$ be the Nielsen and the Reideimeister coincidence numbers
respectively. In this note, we relate $R(f_1,...,f_k)$ with
$R(f_1,f_2),...,R(f_1,f_k)$. When $N$ is a torus or a nilmanifold, we compute
$R(f_1,...,f_k)$ which, in these cases, is equal to $N(f_1,...,f_k)$.
