Question

Five balls of different colours are to be placed in three different boxes such that any box contains at least one ball. What is the maximum number of different ways in which this can be done?

• A. $90.0$
• B. $120.0$
• C. $150.0$
• D. $180.0$

Answer 1

The correct option is C $150.0$

According to the question, we have $5$ balls to be placed in $3$ boxes where no box remains emptyHence, we can have the following kinds of distribution firstly, where the distribution will be $(3,1,1)$ that is, one box gets three balls and the remaining two boxes get one ball each.

No. of ways to choose the box which gets $3$ balls $=3$ (since $3$ boxes are there).

No. of ways to select $3$ balls out of $5$ is = ${5_C}_3$.

No. of ways to distribute the rest of the balls $=2.$

Total no. of ways $=60$

Secondly, where the distribution will be $(1,2,2)$ that is, one box gets one ball and the remaining two boxes get two balls each.

No. of ways to select $1$ balls out of $5$ is = ${5_C}_1$.

No. of ways to choose the box which gets $1$ balls $=3$ (since $3$ boxes are there).

No. of ways to select $2$ balls out of remaining $4$ to go in second box is = ${4_C}_2$.

Total no. of ways $=90$

Total:$60+90=150$.