Question

Five different balls are to be placed in $5$ different boxes randomly. If each box can hold any number of balls, the number of was in which exactly $2$ boxes remain empty is?

• A. $1200$
• B. $1600$
• C. $1400$
• D. $1500$

Answer 1

The correct option is D $1500$

Let $T_3$ is number of ways to fill $5$ Marbles into $3$ boxes such that no box is Empty. So

$T_3=3^5-[{{}^{3}C}_1\times T_2+{{}^{3}C}_2\times T_1]$

where $T_2$ is Number of ways to fill $5$ Marbles into $2$ boxes such that no box is empty and $T_1$ is number of ways to fill $5$ marbles into one box such that it is not empty. Trivially $T_1=1.$So

$T_2=25-[{{}^{2}C}_1\times T_1]$

where So $T_2=30$ and Hence $T_3=243-93=150$

Hence Total ways such that Exactly two are Empty is

${{}^{5}C}_3\times 150=1500$