Question

$5$ balls are to be placed in $4$ boxes. Each box can hold all $5$ balls. In how many different ways can we place the balls so that no box remains empty? If balls and boxes are different

Answer 1

Given $5$ balls $4$ boxes.

But it should be $5$ balls and $3$ boxes.

It can be distributed as $(3,1,1)$ or $(1,2,2)$

Number of ways of selecting $(3,1,1)$

${=}^5{C}_3\times 2\times 3=60$

Number of ways of selecting $(1,2,2)$

${=}^5{C}_1{\times }^4{C}_2\times 3=90$

Total number of ways $=60+90=150$ Ways