On a Functional of the Number of Nonoverlapping Chains Appearing in the Polynomial Scheme and Its Connection with Entropy

Consider $n$ independent chains consisting of $k$ independent polynomial trials with $M$ outcomes. It is assumed that $n, k \to \infty$ and $\ln(n/M^k)=o(k)$.
We find the asymptotics of the normalized logarithm of the number of appearing chains and indicate the connection between this functional and the entropy.