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6.Permutation and Combination
normal
The number of values of $'r'$ satisfying $^{69}C_{3r-1} - ^{69}C_{r^2}=^{69}C_{r^2-1} - ^{69}C_{3r}$ is :-
A
$1$
B
$2$
C
$3$
D
$7$
Solution
${\,^{69}}{{\rm{C}}_{3r – 1}} + {\,^{69}}{{\rm{C}}_{35}} = {\,^{69}}{{\rm{C}}_{{r^2}}} + {\,^{69}}{{\rm{C}}_{{r^2} – 1}}$
$ \Rightarrow {\,^{70}}{C_{3r}} = {\,^{70}}{C_{{r^2}}}$
$\Rightarrow \mathrm{r}^{2}=3 \mathrm{r} \quad$ or $\quad \mathrm{r}^{2}+3 \mathrm{r}=70$
$r = 0{\rm{ or }}\,\,\,r = 3,\,\,\,\,\,\,{r^2} + 3r – 70 = 0$
$ \Rightarrow (r + 10)(r – 7) = 0$
$r = 0,{\rm{ or }}\,\,\,r = 3,\,\,\,\,\,\,\,\,r = – 10,{\rm{ or }}r = 7$
$(r=0, r=-10 \text { not possible })$
Standard 11
Mathematics
