First slide

Binomial theorem for positive integral Index

Question

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

Moderate

A

1
B

2
C

3
D

7

Solution

$^{69} C_{3 r - 1} +^{69} C_{3 r} =^{69} C_{r^{2} - 1} +^{69} C_{r^{2}}$
or $^{70} C_{3 r} =^{70} C_{r^{2}}$
Thus, $r^{2} = 3 r or 70 = 3 r + r^{2} r = 0, 3 or r= 7, - 10$
Hence, r = 3 and 7 (as the given equation is not defined for r = 0 and -10).

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