For integers n and r, let $\left(\begin{array}{c}\mathrm{n}\\ \mathrm{r}\end{array}\right)=\left\{\begin{array}{ll}{}^{\mathrm{n}}\mathrm{C}_{\mathrm{r}},& \mathrm{i}\text{⨍ n≥r≥0}\\ 0,& \mathrm{otherwise}\end{array}\right.$

The maximum value of k for which the sum

$\sum _{\mathrm{i}=0}^{\mathrm{k}+1}\left(\begin{array}{c}10\\ \mathrm{i}\end{array}\right)\left(\begin{array}{c}15\\ \mathrm{k}-\mathrm{i}\end{array}\right)+\sum _{\mathrm{i}=0}^{\mathrm{k}+1}\left(\begin{array}{c}12\\ \mathrm{i}\end{array}\right)\left(\begin{array}{c}13\\ \mathrm{k}+1-\mathrm{i}\end{array}\right)$ exists, is equal to $\mathrm{\_\_\_\_}$

$\begin{array}{rcl}{(1+\mathrm{x})}^{10}& =& {}^{10}{\mathrm{C}}_0+{}^{10}{\mathrm{C}}_1\mathrm{x}+.....+{}^{10}{\mathrm{C}}_{10}{\mathrm{x}}^{10}\end{array}$

$\begin{array}{rcl}{(1+\mathrm{x})}^{15}& =& {}^{15}{\mathrm{C}}_0+{}^{15}{\mathrm{C}}_1\mathrm{x}+.....+{}^{15}{\mathrm{C}}_{15}{\mathrm{x}}^{15}\end{array}$

$\begin{array}{rcl} & & \\ \sum _{\mathrm{i}=0}^{\mathrm{k}} \left({}^{10}{\mathrm{C}}_{\mathrm{i}}\right) \left({}^{15}{\mathrm{C}}_{\mathrm{k}-\mathrm{i}}\right)& =& {}^{10}{\mathrm{C}}_0 {}^{15}{\mathrm{C}}_{\mathrm{k}}+{}^{10}{\mathrm{C}}_1 {}^{15}{\mathrm{C}}_{\mathrm{k}-1}+......+{}^{10}{\mathrm{C}}_{\mathrm{k}} {}^{15}{\mathrm{C}}_0\end{array}$

$\begin{array}{rcl}\sum _{\mathrm{i}=0}^{\mathrm{k}} \left({}^{10}{\mathrm{C}}_{\mathrm{i}}\right) \left({}^{15}{\mathrm{C}}_{\mathrm{k}-\mathrm{i}}\right)& =& \end{array}$Coefficient of ${\mathrm{x}}^{\mathrm{k}}$ in ${(1+\mathrm{x})}^{25}$ 

$\sum _{\mathrm{i}=0}^{\mathrm{k}+1}\left(\begin{array}{c}12\\ \mathrm{i}\end{array}\right)\left(\begin{array}{c}13\\ \mathrm{k}+1-\mathrm{i}\end{array}\right)=$Coefficient of ${\mathrm{x}}^{\mathrm{k}+1}$ in ${(1+\mathrm{x})}^{25}$ 

$\sum _{\mathrm{i}=0}^{\mathrm{k}+1}\left(\begin{array}{c}10\\ \mathrm{i}\end{array}\right)\left(\begin{array}{c}15\\ \mathrm{k}-\mathrm{i}\end{array}\right)+\sum _{\mathrm{i}=0}^{\mathrm{k}+1}\left(\begin{array}{c}12\\ \mathrm{i}\end{array}\right)\left(\begin{array}{c}13\\ \mathrm{k}+1-\mathrm{i}\end{array}\right)$=${}^{25}\mathrm{C}_{\mathrm{k}}+{}^{25}\mathrm{C}_{\mathrm{k}+1}={}^{26}\mathrm{C}_{\mathrm{k}+1}$

$\therefore$k can be as large as possible for 25