Question

If ${}^{n+1}{C}_{r+1}{:}^n{C}_r{:}^{n-1}{C}_{r-1}=11:6:3$, then find the values of $n$ and $r$.

• A. $n=10,r=5$
• B. $n=9,r=4$
• C. $n=11,r=5$
• D. $n=10,r=4$

Answer 1

The correct option is A $n=10,r=5$

Simplifying , we get
$6{}^{n+1}{C}_{r+1}=11{}^n{C}_r$
$\Rightarrow 6(n+1)=11(r+1)$
$\Rightarrow 6n-11r=5$ ... (i)
Also
$3{}^n{C}_r=6{}^{n-1}{C}_{r-1}$
$\Rightarrow 3n=6r$ ...(ii)
Substituting in 1, we get
$12r-11r=5$
$\Rightarrow r=5$
Hence
$n=10$