The number of values of $r$ satisfying the equation $^{69} C_{3 r - 1} -^{69} C_{r^{2}} =^{69} C_{r^{2} - 1} -^{69} C_{3 r}$ is:

1

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2

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4

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7

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Solution

The correct option is B $2$
Solution:
$^{69} C_{3 r - 1} -^{69} C_{r^{2}} =^{69} C_{r^{2} - 1} -^{69} C_{3 r}$
or, $^{69} C_{3 r - 1} +^{69} C_{3 r} =^{69} C_{r^{2}} +^{69} C_{r^{2} - 1}$
or, $^{70} C_{3 r} =^{70} C_{r^{2}}$
Here, $3 r = r^{2}$ or $3 r + r^{2} = 70$ $(∵^{n} C_{r} =^{n} C s$ then $r = s$ or $r + s = n)$
or, $3 r = r^{2}$
or, $r (r - 3) = 0$
or, $r = 0$ which is not possible.
or $r = 3$
$r^{2} + 3 r - 70 = 0$
or, $(r - 7) (r + 10) = 0$
or, $r = 7$ or $r = - 10$ which is not possible.
The value of $r$ satisfying the equation $= 3, 7$
Number of values of $r$ satisfying the equation $= 2$
Hence, B is the correct option.

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