Question

Find the value of $n$ and $r$ in:
${}^n{C}_{r-1}{:}^n{C}_r{:}^n{C}_{r+1}=20:35:42$

Answer 1

$n_{C_{r-1}}:n_{C_r}:n_{C_{r+1}}=20:35:42$

$n_{C_{r-1}}:n_{C_r}=20:35$

$\therefore$ $\frac{n!}{(n-r)!r!}\times \frac{(r-1)!(n-r+1)!}{n!}=35/20$

$\Rightarrow \frac{(n-r+1)}{r}=\frac{7}{4}$

$\Rightarrow 4n-4r+4=7r\Rightarrow 4n+4=11r$

$n_{C_r}:n_{C_{r+1}}=35:42$

$\Rightarrow \frac{r+1}{n-r}=\frac{5}{6}$

$\Rightarrow 6r+6=5n-5r\Rightarrow 5n-11r-6=0$

$\Rightarrow 4n+4=5n-6$

$\Rightarrow n=10$

$\Rightarrow r=4$