Permutations and Combinations 4 Question 4
4. The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball, is
(2012)
(a) 75
(b) 150
(c) 210
(d) 243
Show Answer
Answer:
Correct Answer: 4. (b)
Solution:
- Objects
Distinct
Groups
Objects
Identical
Groups
Description of Situation Here, 5 distinct balls are distributed amongst 3 persons so that each gets at least one ball. i.e. Distinct $\rightarrow$ Distinct So, we should make cases Case I A B C Case II A $\begin{array}{ccc}\text { B } & \text { C } \ 1 & 2 & 2\end{array}$
Number of ways to distribute 5 balls
$$ \begin{aligned} & ={ }^{5} C _1 \cdot{ }^{4} C _1 \cdot{ }^{3} C _3 \times \frac{3 !}{2 !}+{ }^{5} C _1 \cdot{ }^{4} C _2 \cdot{ }^{2} C _2 \times \frac{3 !}{2 !} \\ & =60+90=150 \end{aligned} $$
