For integers n and r, let ( array ll n r array ) = \ cases Cr n , &...

$( 1 + x{)}^{10} = C0 10 + C1 10 x + C2 10 {x}^{2} + \ldots \ldots + C10 10 {x}^{10}$
$( 1 + x{)}^{15} = C0 15 + C1 15 x + \ldots . Ck - 1 15 {x}^{k - 1} + Ck 15 {x}^{k} + Ck + 1 15 {x}^{k + 1} + \ldots {\ldots}^{15}{C}_{15}{x}^{15}$
\mathop{∑}_{i = 0}^{k}\left($\begin{array}{ll}10 \\ i\end{array}$\right)\left($\begin{array}{ll}15 \\ k - i\end{array}$\right) = C0 10 . Ck 15 + C1 10 . Ck - 1 15 + \ldots . + Ck 10 . C0 15
Coefficient of ${x}^{k}$ in $( 1 + x{)}^{25}$
$= Ck 25$
\mathop{∑}_{i = 0}^{k + 1}\left($\begin{array}{ll}12 \\ i\end{array}$\right)\left($\begin{array}{ll}13 \\ k + 1 - i\end{array}$\right) = \mathop{∑}_{i = 0}^{k + 1}\left(Ci 12\right)\left(Ck + 1 - i 13\right) =$C0 12 . Ck + 1 13 + C1 12 . Ck 13 + \ldots . . + Ck + 1 12 . C0 13$

Coefficient of ${x}^{k + 1}$ in $( 1 + x{)}^{25}$
$= Ck + 1 25$

$Ck 25 + Ck + 1 25 = Ck + 1 26$
For maximum value
$k + 1 = 13$
$k = 12$

as $Cr n$ is defined for all values of $n$ as well as $r$ so $Ck + 1 26$ always exists.

Note- Question updated, it was a bonus question in JEE Mains.