Question

The total number of ways in which $5$ balls of different colors can be distributed among $3$ persons so that each person gets at least one ball, is equal to?

• A. $75$
• B. $150$
• C. $210$
• D. $243$

Answer 1

The correct option is B

$150$

Explanation for the correct option:

Find the required number of distribution:

In the question, it is given that we have to distribute $5$ different balls among $3$ persons such that every person gets at least one ball.

Since, every ball has three options. Therefore, the total number of distributions without restrictions are $3^5$.

Now, the number of ways in which one person gets all the balls are $3$.

And the number of ways in which one person gets no-ball are $3·\left(2^5-2\right)=90$.

So, the total number of required distribution can be given by $3^5-90-3=150$.

Therefore, the total number of ways in which $5$ balls of different colors can be distributed among $3$ persons so that each person gets at least one ball is equal to $150$.

Hence, option (B) is the correct answer.