For integers $n$ and $r$, let $\binom{n}{r}= \begin{cases}{ }^n C_r, & \text { if } n \geq r \geq 0 \\ 0, & \text { otherwise }\end{cases}$ The maximum value of $k$ for which the sum $\sum_{\mathrm{i}=0}^{\mathrm{k}}\binom{10}{\mathrm{i}}\binom{15}{\mathrm{k}-\mathrm{i}}+\sum_{\mathrm{i}=0}^{\mathrm{k}+1}\binom{12}{\mathrm{i}}\binom{13}{\mathrm{k}+1-\mathrm{i}}$ exists, is equal to