Question

The number of values of $r$ satisfying the equation $^{39}C_{3r-1} - ^{39}C_{r^{2}} = ^{39}C_{r^{2}-1} - ^{39}C_{3r} $ is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The Correct Option is B

$^{39}C_{3r-1} - ^{39}C_{r^{2}} = ^{39}C_{r^{2}-1} - ^{39}C_{3r} $
$ \Rightarrow ^{39} C_{3r-1} + ^{39}C_{3r} = ^{39}C_{r^{2}-1} + ^{39}C_{r^{2}} $
$\Rightarrow ^{40}C_{3r} = ^{40}C_{r^{2}} $
$\Rightarrow r^{2} = 3r $ or $r^{2} = 40 - 3r $
$\Rightarrow r=0 ,3 $ or $ 8,5$
$3$ and $5$ are the values as the given equation is not defined by $r = 0$
and $r = -8$.
Hence, the number of values of $r$ is $2$.