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For integers \( n \) and \( r \), let \( \left(\begin{array}{l}n \\ r\end{array}\right)=\left\{\begin{array}{ll}{ }^{n} C_{r}, & \text { if } n \geq r \geq 0 \\ 0, & \text { otherwise }\end{array}\right. \)
D
The maximum value of \( k \) for which the sum
\( \sum_{i=0}^{k}\left(\begin{array}{c}10 \\ i\end{array}\right)\left(\begin{array}{c}15 \\ k-i\end{array}\right)+\sum_{i=0}^{\mathrm{K}+1}\left(\begin{array}{c}12 \\ i\end{array}\right)\left(\begin{array}{c}13 \\ k+1-i\end{array}\right) \) exists, is equal to
Note: NTA has dropped this question in the final official answer key.
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