# multinomial sum upper bound

I am looking for an upper bound on the following sum, where $$p_1,\dots,p_r > 0$$ and $$\sum_i p_i\leq 1$$.

$$\sum_{0\leq i_1,\dots,i_r\leq n} \binom{i_1+\cdots+i_r}{i_1,\dots,i_r} p_{1}^{i_1}\cdots p_{r}^{i_r}$$

I tried the upper bound in MSE 2589433 for the multinomial coefficient but I had no idea how to control things like $$\exp\left(i_1\log\frac{i_1+\cdots+i_r}{i_1}\right)$$ in a nice way. A trivial bound such as $$\log\frac{i_1+\cdots+i_r}{i_1}\leq \log\frac{rn}{i_1} \leq \log(rn)$$ is too cruel.

Note that $$\sum_{0\leq i_1,\dots,i_r\leq n} \binom{i_1+\cdots+i_r}{i_1,\dots,i_r} p_{1}^{i_1}\cdots p_{r}^{i_r}\leq\sum_{0\leq i_1,i_2,\ldots,i_r\atop\sum i_j\leq nr }\binom{i_1+\cdots+i_r}{i_1,\dots,i_r} p_{1}^{i_1}\cdots p_{r}^{i_r}=\\ \sum_{l=0}^{nr}\sum_{i_1+\ldots+i_r=l}\binom{i_1+\cdots+i_r}{i_1,\dots,i_r} p_{1}^{i_1}\cdots p_{r}^{i_r}=\sum_{l=0}^{nr}(p_1+\ldots+p_r)^l$$ This expresion is always $$\leq nr+1$$ and for $$p_1+p_2+\ldots+p_r<1$$ we have also the better estimate $$\leq \frac{1-(p_1+\ldots+p_r)^{nr+1}}{1-(p_1+\ldots+p_r)}$$