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  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

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Q. 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

IIT JEEIIT JEE 2012Permutations and Combinations

Solution:

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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
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$=\left({ }^{5} C_{1} \cdot{ }^{4} C_{1} \cdot{ }^{3} C_{3} \times \frac{3 !}{2 !}\right)+\left({ }^{5} C_{1} \cdot{ }^{4} C_{2} \cdot{ }^{2} C_{2} \times \frac{3 !}{2 !}\right)$
$=60+90=150$