 Solution

On an $n$-sample, each coordinate partition has at most $(n+1{)}^{m-1}$ placements, hence all $p$ partitions have at most $(n+1{)}^{p(m-1)}$ configurations. For each fixed configuration, the sign class from a vector space of dimension $1+p(m-1)$ has at most $(n+1{)}^{1+p(m-1)}$ patterns. Multiplication gives