For integers $\mathrm{n}$ and $\mathrm{r}$, let $\left(\begin{array}{l}\mathrm{n} \\ \mathrm{r}\end{array}\right)=\left\{\begin{array}{cc}{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}, & \text { if } \mathrm{n} \geq \mathrm{r} \geq 0 \\ 0, & \text { otherwise }\end{array}\right.$
The maximum value of $\mathrm{k}$ for which the sum
$\sum_{i=0}^{k}\left(\begin{array}{c}10 \\ i\end{array}\right)\left(\begin{array}{c}15 \\ k-i\end{array}\right)+\sum_{i=0}^{k+1}\left(\begin{array}{c}12 \\ i\end{array}\right)\left(\begin{array}{c}13 \\ k+1-i\end{array}\right)$ exists, is
equal to
$\sum_{i=0}^{k}\left(\begin{array}{c}10 \\ i\end{array}\right)\left(\begin{array}{c}15 \\ k-i\end{array}\right)+\sum_{i=0}^{k+1}\left(\begin{array}{c}12 \\ i\end{array}\right)\left(\begin{array}{c}13 \\ k+1-i\end{array}\right)$
${ }^{25} \mathrm{C}_{\mathrm{k}}+{ }^{25} \mathrm{C}_{\mathrm{k}+1}$
${ }^{26} \mathrm{C}_{k+1}$
as ${ }^{n} C_{r}$ is defined for all values of $n$ as will as r so ${ }^{26} \mathrm{C}_{\mathrm{k}+1}$ always exists
Now $\mathrm{k}$ is unbounded so maximum value is not defined.