Q79 / 125 · BITSAT 2016 · Mathematics
The number of values of satisfying the equation ${ }^{39} \mathrm{C}_{3 \mathrm{r}-1}-{ }^{39} \mathrm{C}_{\mathrm{r}^{2}}={ }^{39} \mathrm{C}_{\mathrm{r}^{2}-1}-{ }^{39} \mathrm{C}_{3 \mathrm{r}}$ is
- A1
- Correct answerB2
- C3
- D4
Solution
${ }^{39} \mathrm{C}_{3 \mathrm{r}-1}-{ }^{39} \mathrm{C}_{\mathrm{r}^{2}}={ }^{39} \mathrm{C}_{\mathrm{r}^{2}-1}-{ }^{39} \mathrm{C}_{3 \mathrm{r}}$
$\Rightarrow{ }^{39} \mathrm{C}_{3 \mathrm{r}-1}+{ }^{39} \mathrm{C}_{3 \mathrm{r}}={ }^{39} \mathrm{C}_{\mathrm{r}^{2}-1}+{ }^{39} \mathrm{C}_{\mathrm{r}^{2}}$
$\Rightarrow{ }^{40} \mathrm{C}_{3 \mathrm{r}}={ }^{40} \mathrm{C}_{\mathrm{r}^{2}}$
$\Rightarrow \mathrm{r}^{2}=3 \mathrm{r}$ or $\mathrm{r}^{2}=40-3 \mathrm{r}$
$\Rightarrow \mathrm{r}=0,3$ or -8,5
3 and 5 are the values as the given equation is not defined by $r=0$ and $r=-8$. Hence, the number of values of $\mathrm{r}$ is 2 .
Hence, the correct answer is option: 'B'.