# For integers n and r, let $\binom{n}{r}=\left\{\begin{matrix} ^{n}C_{r} & \text{if} \; \geq r\geq 0\\ 0,& \text{otherwise} \end{matrix}\right.$ The maximum value of k for which the sum  $\sum_{i=0}^{k}\binom{10}{i}\binom{15}{k-i}+\sum_{i=0}^{k+1}\binom{12}{i}\binom{13}{k+1-i}$ exists, is equal to _____ Option: 1 Not defined Option: 2 25 Option: 3 26 Option: 4 15

$\begin{array}{l} \sum_{i=0}^{k}\left(\begin{array}{c} 10 \\ i \end{array}\right)\left(\begin{array}{c} 15 \\ k-i \end{array}\right)+\sum_{i=0}^{k+1}\left(\begin{array}{c} 12 \\ i \end{array}\right)\left(\begin{array}{c} 13 \\ k+1-i \end{array}\right) \\\\ { }^{25} C_{k}+{ }^{25} C_{k+1} \\\\ { }^{26} C_{k+1} \end{array}$

as ${ }^{n} C_{r}$ is defined for all values of n as will as r

so ${ }^{26} \mathrm{C}_{\mathrm{k}+1}$ always exists

Now k is unbounded so maximum value is not defined.

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