Question

If ${}^n{C}_x={}^n{C}_y$ and $n\ne y$ then prove that $x+y=n.$

Answer 1

We have

${}^n{C}_x={}^n{C}_y={}^n{C}_{x-y}[\because {}^n{C}_r={}^n{C}_{x-r}]$

$\Rightarrow x=yorx=n-y$

$\Rightarrow x=n-y[\because x\ne y(given\left)\right]$

$\Rightarrow x+y=n.$

Hence, ${}^n{C}_x={}^n{C}_y$ and $x\ne y$ then $x+y=n.$