Question ID: #1562
English: $~^{n-1}C_{r} = (k^{2}-8) ~^{n}C_{r+1}$ if and only if:
(A) $2\sqrt{2}\lt k\le3$ (B) $2\sqrt{3}\lt k\le3\sqrt{2}$ (C) $2\sqrt{3}\lt k<3\lt \sqrt{3}$ (D) $2\sqrt{2}\lt k\lt 2\sqrt{3}$
Hindi: $~^{n-1}C_{r} = (k^{2}-8) ~^{n}C_{r+1}$ यदि और केवल यदि:
(A) $2\sqrt{2}\lt k\le3$ (B) $2\sqrt{3}\lt k\le3\sqrt{2}$ (C) $2\sqrt{3}\lt k\lt 3\sqrt{3}$ (D) $2\sqrt{2}\lt k\lt 2\sqrt{3}$
Gujarati: $~^{n-1}C_{r} = (k^{2}-8) ~^{n}C_{r+1}$ જો અને માત્ર જો:
(A) $2\sqrt{2}\lt k\le3$ (B) $2\sqrt{3}\lt k\le3\sqrt{2}$ (C) $2\sqrt{3}\lt k\lt 3\sqrt{3}$ (D) $2\sqrt{2}\lt k\lt 2\sqrt{3}$
Hint:
- English: Use the property $\frac{^{n-1}C_r}{^nC_{r+1}} = \frac{r+1}{n}$ and the boundary condition $0 < \frac{r+1}{n} \le 1$ to establish inequalities for $k$.
- Hindi: $k$ का परिसर ज्ञात करने के लिए गुणधर्म $\frac{^{n-1}C_r}{^nC_{r+1}} = \frac{r+1}{n}$ और शर्त $0 < \frac{r+1}{n} \le 1$ का उपयोग करें।
- Gujarati: $k$ નો વિસ્તાર શોધવા માટે ગુણધર્મ $\frac{^{n-1}C_r}{^nC_{r+1}} = \frac{r+1}{n}$ અને શરત $0 < \frac{r+1}{n} \le 1$ નો ઉપયોગ કરો.
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