Question

The total number of ways in which $5$ balls of different colours can be distributed among $3$ persons so that each person gets atleast one ball is

• A. $150$
• B. $160$
• C. $36$
• D. $144$

Answer 1

Answer: (A)

$150$

We have,

Total number of balls $=5$

Total no. of persons $=3$

According each person should get at least one ball.

$\therefore$The selection of balls can be done by

$(2,2,1)$ or$(1,1,3)$

$i.e,\left({}^5{C}_2{\times }^3{C}_2{\times }^1{C}_1\right)or\left({}^5{C}_1{\times }^4{C}_1{\times }^3{C}_3\right)$

$=(10\times 3\times 1)or(5\times 4\times 1)$

$=30or20ways.$

Then,

$(2,2,1)$ selection is distributed in $\frac{3!}{2!}=3ways$.

$(1,1,3)$ selection is distributed in $\frac{3!}{2!}=3ways$.

Then, the required number of ways

$=(30\times 3)+(20\times 3)$

$=90+60$

$=150$

Hence, this is the answer.