Question

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.

Answer 1

The bag contains $12$ two rupee coins, $7$ one rupee coins and $4$ half a rupee coins.

Total coins in the bag $=12+7+4=23$

Selecting three coins from the bag $={}^{23}{C}_3=1771$

Sum of three coins will be maximum when all the three coins selected are of rupee two.

Therefore,

Selecting three coins of rupee two $={}^{12}{C}_3=220$

Probability of selecting three coins with maximum sum $=\frac{220}{1771}=\frac{20}{161}$

Sum of three coins will be minimum when all the three coins selected are of half a rupee coin.

Therefore,

Selecting three coins of half a rupee $={}^4{C}_3=4$

Probability of selecting three coins with minimum sum $=\frac{4}{1771}$

Selecting three different coins $={12}_1^C\times {}^7{C}_1\times {}^4{C}_1=12\times 7\times 4=336$

Probability of selecting three different coins from the bag $=\frac{336}{1771}=\frac{48}{253}$