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

Answer: (B)

$150$

Answer 2

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 $\rightarrow$ Distinct So, we should make cases Case I \begin {array} \ A \\\ 1 \end {array} \begin {array} \ B \\\ 1 \end {array}\begin {array} \ C \\\ 2 \end {array} \bigg \\} Case II \begin {array} \ A \\\ 1 \end {array} \begin {array} \ B \\\ 2 \end {array}\begin {array} \ C \\\ 2 \end {array} \bigg \\} Number of ways to distribute $5$ balls =$\bigg( ^5C_1.\ ^4C_1.\ ^3C_3 \times \ \frac{3!}{2!}\bigg)+\bigg( ^5C_1.\ ^4C_2.\ ^2C_2 \times \ \frac{3!}{2!}\bigg)$ $=60+90=150$