Repetitions of multinomial coefficients and a generalization of Singmaster's conjecture

Given two integers $k\ge 2$ and $a>1$, let $N_k(a)$ stand for the number of multinomial coefficients, with $k$ terms, equal to $a$. We study the behavior of $N_k(a)$ and show that its average and normal orders are equal to $k(k-1)$. We also prove that $N_k(a)=O((\log a\mathrm{/}\log \log a{)}^{k-1})$ and make several propositions about extreme results regarding large values of $N_k(a)$.