For integers $n$ and $r$, let $\left(\begin{array}{l}n \\ r\end{array}\right)= \begin{cases}{ }^{n} C_{r}, & \text { if } n \geq r \geq 0 \\ 0, & \text { otherwise }\end{cases}$
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
Note: NTA has dropped this question in the final official answer key.
$(1+x)^{10}={ }^{10} C_{0}+{ }^{10} C_{1} x+{ }^{10} C_{2} x^{2}+\ldots \ldots+{ }^{10} C_{10} x^{10}$
$(1+x)^{15}={ }^{15} C_{0}+{ }^{15} C_{1} x+\ldots{ }^{15} C_{k-1} x^{k-1}+{ }^{15} C_{k} x^{k}+{ }^{15} C_{k+1} x^{k+1}+\ldots \ldots{ }^{15} C_{15} x^{15}$
$\sum_{i=0}^{k}\left(10 C_{i}\right)\left(15 C_{k-1}\right)={ }^{10} C_{0} \cdot{ }^{15} C_{k}+{ }^{10} C_{1} \cdot{ }^{15} C_{k-1}+\ldots \ldots+{ }^{10} C_{k} \cdot{ }^{15} C_{0}$
Coefficient of $x_{k}$ in $(1+x)^{25}$
$={ }^{25} \mathrm{C}_{\mathrm{k}}$
$\sum_{i=0}^{k+1}\left({ }^{12} C_{i}\right)\left({ }^{13} C_{k+1-i}\right)={ }^{12} C_{0} \cdot{ }^{13} C_{k+1}+{ }^{12} C_{1} \cdot{ }^{13} C_{k}+\ldots \ldots+{ }^{12} C_{k+1} \cdot{ }^{13} C_{0}$
Coefficient of $x^{k+1}$ in $(1+x)^{25}$
$={ }^{25} \mathrm{C}_{\mathrm{k}+1}$
${ }^{25} \mathrm{C}_{\mathrm{k}}+{ }^{25} \mathrm{C}_{\mathrm{k}+1}={ }^{26} \mathrm{C}_{\mathrm{k}+1}$
For maximum value
$k+1=13$
$\mathrm{K}=12$
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