The number of value(s) of $r$ satisfying the equation $^{69} C_{3 r - 1} -^{69} C_{r^{2}} =^{69} C_{r^{2} - 1} -^{69} C_{3 r}$ is

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Solution

Given
$^{69} C_{3 r - 1} -^{69} C_{r^{2}} =^{69} C_{r^{2} - 1} -^{69} C_{3 r} \Rightarrow^{69} C_{3 r - 1} +^{69} C_{3 r} =^{69} C_{r^{2}} +^{69} C_{r^{2} - 1} \Rightarrow^{70} C_{3 r} =^{70} C_{r^{2}} \Rightarrow 3 r = r^{2} or 3 r + r^{2} = 70 \Rightarrow r = 0, 3 or r^{2} + 3 r - 70 = 0 \Rightarrow r = 0, 3 or (r + 10) (r - 7) = 0 \Rightarrow r = 0, 3, - 10, 7$

As $3 r - 1, r^{2}, r^{2} - 1, 3 r \in W$ and less than or equal to $69$ , so the required values are
$r = 3, 7$

Hence, the number of values of $r$ is $2.$

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