The number of values of r satisfying ^{69}C_{ | vedclass

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(B) Given equation: $^{69}C_{3r-1} - ^{69}C_{r^2} = ^{69}C_{r^2-1} - ^{69}C_{3r}$ Rearranging the terms: $^{69}C_{3r-1} + ^{69}C_{3r} = ^{69}C_{r^2} +

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$2$

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(B) Given equation: $^{69}C_{3r-1} - ^{69}C_{r^2} = ^{69}C_{r^2-1} - ^{69}C_{3r}$ Rearranging the terms: $^{69}C_{3r-1} + ^{69}C_{3r} = ^{69}C_{r^2} + ^{69}C_{r^2-1}$ Using the identity $^{n}C_{r} + ^{n}C_{r-1} = ^{n+1}C_{r}$: $^{70}C_{3r} = ^{70}C_{r^2}$ This implies either $3r = r^2$ or $3r + r^2 = 70$. Case $1$: $r^2 - 3r = 0$ $\Rightarrow r(r-3) = 0$ $\Rightarrow r = 0$ or $r = 3$. Case $2$: $r^2 + 3r - 70 = 0$ $\Rightarrow (r+10)(r-7) = 0$ $\Rightarrow r = -10$ or $r = 7$. Since $nCr$ is defined for $n \ge r \ge 0$,we check the values: For $r=0$: $^{69}C_{-1}$ is undefined. For $r=-10$: $^{69}C_{-31}$ is undefined. For $r=3$: $^{69}C_{8} - ^{69}C_{9} = ^{69}C_{8} - ^{69}C_{9}$ (Valid). For $r=7$: $^{69}C_{20} - ^{69}C_{49} = ^{69}C_{48} - ^{69}C_{21}$ (Valid,since $^{69}C_{49} = ^{69}C_{20}$ and $^{69}C_{48} = ^{69}C_{21}$). Thus,the valid values for $r$ are $3$ and $7$. The number of values is $2$.

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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