For integers n and r, let (\table n ; r)=\\begin{array}{c}{\end{array}^{n} C_{r^{'}} ,{,} \if n \geq r \geq 0 ; 0 {,}, \otherwise The maximum value of k for which the sum, \sum_{i=0}^{k}(\table 10 ; i)(\table 15 ; k-i)+\sum_{i=0}^{k+1}(\table12 ; i)(\table 13 ; k+1-i) exists, is equal to .......... [24 Feb 2021 Shift 2]