A bag contains 12 two rupee coins, 7 one rupee coins and 4 half a rupee coins. If three coins are selected at random, then find the probability that 

 i) the sum of three coins is maximum

 ii) the sum of three coins is minimum 

iii) each coin is of different value

A bag contains $2Rs$  coins $–12$

1Rs coins $–7$

 $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ Rs coins $–4$

Three coins are selected at random  $\Rightarrow n\left(S\right){=}^{23}{C}_3$

i) Let $E$be the event of getting 3 coins whose sum is max.

$n\left(E\right){=}^{12}{C}_3$  

  $P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}=\frac{{ }^{12}{C}_3}{{ }^{23}{C}_3}$

ii) Let $E$  be the event of three coins whose sum is min.

$n\left(E\right){=}^4{C}_3$    

 $P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}=\frac{{ }^4{C}_3}{{ }^{12}{C}_3}$

iii) $E$ be the event of getting 3 coins, each coin is of different value

$n\left(E\right){=}^{12}{C}_1{\cdot }^7{C}_1{\cdot }^4{C}_1$

$P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}=\frac{{ }^{12}{C}_1{\cdot }^7{C}_1{\cdot }^4{C}_1}{{ }^{12}{C}_3}$