Question

There are two bags each containing m balls. If a man has to select equal number of balls from both the bags the number of ways in which he can do so if he must choose at least one ball from each bag is

• A. $m^2$
• B. ${}^{2m}{C}_m$
• C. ${}^{2m}{C}_m-1$
• D. None of these

Answer 1

Answer: (C)

${}^{2m}{C}_m-1$

The number of ways of selecting $r(0\le r\le m)$ balls out of $m$ is ${}^m{C}_r$. Therefore, the number of ways if selecting $r$ balls from each of the bag is ${(}^m{C}_r{)}^2$. Further the number of ways of selecting equal number of balls from each of two bags, choosing at least one from each bag, is
${(}^m{C}_1{)}^2+{(}^m{C}_2{)}^2+...+{(}^m{C}_m{)}^2{-}^{2m}{C}_m-1$
$[\because {(}^m{C}_0{)}^2+{(}^m{C}_1{)}^2+....+{(}^m{C}_1{)}^2{=}^{2m}{C}_m]$.