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

$\therefore$ Each person gets at lest one ball.
$\therefore$ 3 persons can have 5 balls in the following systems
$\begin{array}{cccc}Person& I& II& III\\ No.ofballs& 1& 1& 3\end{array}or\begin{array}{cccc}Person& I& II& III\\ No.ofballs& 1& 2& 2\end{array}$
The number of ways to distribute the balls in first system
$={}^5{C}_1\times {}^4{C}_1\times {}^3{C}_3$
Also 3, persons having 1,1 and 3 balls can be arranged in $\frac{3!}{2!}$ways.
$={}^5{C}_1\times {}^4{C}_1\times {}^3{C}_3\times \frac{3!}{2!}=60$
Similarly the total no.of ways to distribute 1,2,2 balls to the three perrsons $={}^5{C}_1\times {}^4{C}_2\times {}^2{C}_2\times \frac{3!}{2!}=90$
$\therefore$ The required numbers of ways = 60 + 90 = 150.