Question

If ${}^{n}C{}_{r-1}=36{,}^n{C}_r=84{,}^{n}C{}_{r+1}=126$, then $(n,r)$$=$

• A. $(9,6)$
• B. $(9,5)$
• C. $(9,3)$
• D. $(9,2)$

Answer 1

The correct option is C $(9,3)$

$\frac{{}^n{C}_r}{{}^n{C}_{r-1}}=\frac{n-r+1}{r}=\frac{84}{36}=\frac{7}{3}\Rightarrow 3n-10r+3=0$
and $\frac{{}^n{C}_{r+1}}{{}^n{C}_r}=\frac{n-r}{r+1}=\frac{126}{84}=\frac{3}{2}\Rightarrow 2n-5r-3=0$
Solving both equations for $n$ and $r$,
$(n.r)=(9,3)$