For integers n and r, let ( l n r )= ll ^ n C _ r , & if n r 0 0,… - Vidyadip

MCQ

For integers $n$ and $r$, let $\left(\begin{array}{l} n \\ r \end{array}\right)=\left\{\begin{array}{ll}{ }^{n} C _{ r }, & \text { if } n \geq r \geq 0 \\ 0, & \text { otherwise }\end{array}\right.$The maximum value of $k$ for which the sum $\sum_{i=0}^{k}\left(\begin{array}{c}10 \\ i\end{array}\right)\left(\begin{array}{c}15 \\ k-i\end{array}\right)+\sum_{i=0}^{k+1}\left(\begin{array}{c}12 \\ i\end{array}\right)\left(\begin{array}{c}13 \\ k+1-i\end{array}\right)$ exists, is equal to $...... .$

✓
Not define
B
$24$
C
$36$
D
$20$

✓

Answer

Correct option: A.

Not define

$\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}$
${ }^{26} C _{ k +1}^{ }$
as ${ }^{ n } C _{ r }$ is defined for all values of $n$ as will as $r$ so ${ }^{26} C _{ k +1}$ always exists
Now $k$ is unbounded so maximum value is not defined.

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