For integers \( n \) and \( r \), let \( {n \choose r} = \begin{cases} \,\,^n C_r, & \text{if } n \ge r \ge 0 \\ 0, & \text{otherwise} \end{cases} \)
The maximum value of \( k \) for which the sum \( \displaystyle \sum_{i=0}^{k} {10 \choose i}{15 \choose k-i} + \sum_{i=0}^{k+1} {12 \choose i}{13 \choose k+1-i} \) exists, is equal to ______.
