The value of r for which

Question:

The value of $\mathrm{r}$ for which

${ }^{20} C_{r}{ }^{20} C_{0}+{ }^{20} C_{r-1}{ }^{20} C_{1}+{ }^{20} C_{r-2}{ }^{20} C_{2}+\ldots+{ }^{20} C_{0}{ }^{20} C_{r}$

is maximum, is :

  1. (1) 15

  2. (2) 20

  3. (3) 11

  4. (4) 10


Correct Option: , 2

Solution:

Consider the expression ${ }^{20} C_{r}{ }^{20} C_{0}+{ }^{20} C_{r}-1{ }^{20} C_{1}$ $+{ }^{20} C_{r-2}{ }^{20} C_{2}+\ldots+{ }^{20} C_{0} \cdot{ }^{20} C_{r}$

For maximum value of above expression $r$ should be equal to 20 .

as ${ }^{20} C_{20} \cdot{ }^{20} C_{0}+{ }^{20} C_{19} \cdot{ }^{20} C_{1}+\ldots+{ }^{20} C_{20} \cdot{ }^{20} C_{0}$

$=\left({ }^{20} C_{0}\right)^{2}+\left({ }^{20} C_{1}\right)^{2}+\cdots+\left({ }^{20} C_{20}\right)^{2}={ }^{40} C_{20}$

Which is the maximum value of the expression,

So, $r=20$

Leave a comment

Close

Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now