For integers n and r, let n r = {^n C_r, if n ≥ r ≥ 0; 0, otherwise} The maximum value of k for which the sum Σ_i=0^k 10 i 15 k-i +Σ_i=0^k+1 12 i 13…

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