The number of values of 'r' satisfyin | vedclass

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Question

${\,^{69}}{{\rm{C}}_{3r - 1}} + {\,^{69}}{{\rm{C}}_{35}} = {\,^{69}}{{\rm{C}}_{{r^2}}} + {\,^{69}}{{\rm{C}}_{{r^2} - 1}}$

$ \Rightarrow {\,^{70}}{C

Vedclass · Accepted Answer

$2$

Vedclass · Answer

${\,^{69}}{{m{C}}_{3r - 1}} + {\,^{69}}{{m{C}}_{35}} = {\,^{69}}{{m{C}}_{{r^2}}} + {\,^{69}}{{m{C}}_{{r^2} - 1}}$

$ \Rightarrow {\,^{70}}{C_{3r}} = {\,^{70}}{C_{{r^2}}}$

$\Rightarrow \mathrm{r}^{2}=3 \mathrm{r} \quad$     or      $\quad \mathrm{r}^{2}+3 \mathrm{r}=70$

$r = 0{m{ or }}\,\,\,r = 3,\,\,\,\,\,\,{r^2} + 3r - 70 = 0$

$ \Rightarrow (r + 10)(r - 7) = 0$

$r = 0,{m{ or }}\,\,\,r = 3,\,\,\,\,\,\,\,\,r =  - 10,{m{ or }}r = 7$

$(r=0, r=-10 	ext { not possible })$

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

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