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Q.
The number of values of $r$ satisfying the equation ${ }^{39} C_{3 r-1}-{ }^{39} C_{r^{2}}={ }^{39} C_{r^{2}-1}-{ }^{39} C_{3 r}$ is
Permutations and Combinations
Solution:
${ }^{39} C_{3 r-1}-{ }^{39} C_{r^{2}}={ }^{39} C_{r^{2}-1}-{ }^{39} C_{3 r}$
$\Rightarrow { }^{39} C_{3 r-1}+{ }^{39} C_{3 r}={ }^{39} C_{r^{2}-1}+{ }^{39} C_{r^{2}} $
$\Rightarrow { }^{40} C_{3 r}={ }^{40} C_{r^{2}}$
$\Rightarrow r^{2}=3 r$ or $r^{2}=40-3 r$
$ \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 .