Question

If ${}^{15}{C}_r{:}^{15}{C}_{r-1}=11:5$, find r.

Answer 1

$\frac{{}^{15}{C}_r}{{}^{15}{C}_{r-1}}=\frac{11}{5}$

$\Rightarrow \frac{\frac{15!}{r!(15-r)!}}{\frac{15!}{(r-1)!(15-r+1)}}=\frac{11}{5}$

$\Rightarrow \frac{(r-1)!\cdot 15(15-r+1)!}{r!(15-r)!}=\frac{11}{5}$

$\Rightarrow \frac{15-r+1}{r}=\frac{11}{5}$

$\Rightarrow 75-5r+5=11r$

$\Rightarrow 80=16r$

$\Rightarrow r=5$