Summary: Fix positive integers \(k \geq 2\), \(j \geq 2\) and numbers \(p_1, p_2, \ldots, p_k\) such that \(0 < p_i < 1\) for all \(i = 1, 2, \ldots, k\), and \(\sum_{i = 1}^k p_i = 1\). For a positive integer \(n\), let \[ b_{n, j, k}(p_1, p_2, \ldots, p_k) \equiv \sum_{(n_1, n_2, \ldots, n_k) \in T_{n, k}}(\frac{n!}{n_1! n_2! \cdots n_k!} p_1^{n_1} p_2^{n_2} \cdots p_k^{n_k})^j, \] where \(T_{n, k}\) is the set \(\{(n_1, n_2, \ldots, n_k) : n_i \in \{0, 1, 2, \ldots, n \}, \sum_{i = 1}^k n_i = n \}\). Then there exists \(0 < b_{j, k}(p_1, p_2, \ldots, p_k) < \infty\) such that \[ n^{(j - 1)(k - 1) / 2} b_{n, j, k}(p_1, p_2, \ldots, p_k) \to b_{j, k}(p_1, p_2, \ldots, p_k)\,\,\,\,\,\,\, (1) \] as \(n \to \infty\).