If
Question:

If $\sum_{r=0}^{25}\left\{{ }^{50} C_{r} \cdot{ }^{50-r} C_{25-r}\right\}=K\left({ }^{50} C_{25}\right)$, then $K$ is equal to:

  1. (1) $(25)^{2}$

  2. (2) $2^{25}-1$

  3. (3) $2^{24}$

  4. (4) $2^{25}$


Correct Option: , 4

Solution:

$\sum_{r=0}^{25}\left({ }^{50} C_{r} \cdot{ }^{50-r} C_{25-r}\right)=\sum_{r=0}^{25}\left(\frac{\lfloor 50}{|50-r| r} \frac{\mid 50-r}{|25| 25-r}\right)$

$=\sum_{r=0}^{25}\left(\frac{\lfloor 50}{\lfloor 25} \times \frac{1}{\lfloor 25} \times\left(\frac{\lfloor 25}{\lfloor 25-r \mid r}\right)\right)$

$={ }^{50} C_{25} \sum_{r=0}^{25}{ }^{25} C_{r}={ }^{50} C_{25}\left(2^{25}\right)$

Then, by comparison, $K=2^{25}$

 

Administrator

Leave a comment

Please enter comment.
Please enter your name.