Chapter 17, Problem 17.1E

Interpretation Introduction

Interpretation:

The possible ways of putting two balls in three boxes, with one ball in each box is to be drawn.

Concept introduction:

Combination formula is the possible way of grouping of distinguishable objects into various subsystems. The equation for combination formula is as follows:

$C=\frac{N!}{\underset{i=1}{\overset{m}{\prod n_i!}}}$

Where,

• $m$ is the subsystems.

• $N$ is the object.

• $n_i$ is the object in the ith subsystem.

Answer to Problem 17.1E

The possible ways of putting two balls in three boxes, with one ball in each box is shown below.

Explanation of Solution

Combination formula is the possible way of grouping of distinguishable objects into various subsystems. The equation for combination formula is as follows:

$C=\frac{N!}{\underset{i=1}{\overset{m}{\prod n_i!}}}$

Where,

• $N!$ is the $N$ factorial and values can be $1$, $2$, $3$, $N$.

• $\underset{i=1}{\overset{m}{\prod n_i!}}$ is the product of all the $n_i$ values.

It is given that two balls are present in three separate boxes. Therefore, the given value of $N$ is $3$ and total objects are $2$.

Substitute the given values in the above formula.

$\begin{array}{rllll}C& =& \frac{N!}{\underset{i}{\overset{}{\prod n_i!}}}& =& 1m\\ & =& \frac{3!}{2!}\\ & =& \frac{3\times 2!}{2!}\\ & =& 3\end{array}$

Thus, the number of possibilities of putting two balls in three boxes, with one ball in each box is $3$.

Therefore, the possible ways of putting two balls in three boxes, with one ball in each box is shown below.

Figure 1

Conclusion

The possible ways of putting two balls in three boxes, with one ball in each box is shown in Figure 1.