Question

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 ?___

• A. 75
• B. 150
• C. 210
• D. 243

Answer 1

The correct option is B 150

Objects Groups Objects Groups
Distinct Distinct Identical Identical
Distinct Identical Identical Distinct
Description of Situation Here, 5 distinct balls are distributed amongst 3 persons so that each gets at least one ball. i.e. Distinct $\to$ Distinct
So, we should make cases
$CaseI\begin{array}{cccc}A& B& C& \\ 1& 1& 3& \end{array}\}CaseII\begin{array}{cccc}A& B& C& \\ 1& 2& 2& \end{array}\}$
Number of ways to distribute 5 balls
$={(}^5{C}_1{.}^4{C}_1{.}^3{C}_3\times \frac{3!}{2!})+{(}^5{C}_1{.}^4{C}_2{.}^2{C}_2\times \frac{3!}{2!})=60+90=150$