Denote $N :=(n_1,n_2,..., n_s), n_j\in \mathbb{N^\ast}$ and let $k\in \mathbb{N}.$ I want to prove the following inequality
$$\sum_{|M|=k}\Big(^k_M\Big)\left[\Big(^{k+|N|}_{N+M}\Big)\right]^{-2}\leq 1,$$ where $M = (m_1, m_2, ...,m_s)$ and $\Big(^k_M\Big)$ is the multinomial coefficient defined by $$\Big(^k_M\Big) = \frac{k!}{\prod_{k=1}^sm_k!}.$$
The simple case where $s=2$ corresponds to
$$\sum_{m=0}^k\Big(^k_m\Big)\left[\Big(^{k+n_1+n_2}_{n_1+m}\Big)\right]^{-2}\leq 1,$$
which can be done by proving that
$$\Big(^k_m\Big)\left[\Big(^{k+n_1+n_1}_{n_1+m}\Big)\right]^{-1}\leq 1 \quad \text{and} \quad \Big(^{k+n_1+n_1}_{n_1+m}\Big)\geq k+1.$$
In this proof I used the Pascal's triangle...