Question

In how many ways can $mn$ letters be posted in $n$ letter boxes?

1. ${{\left( mn \right)}^{n}}$
2. ${{m}^{mn}}$
3. ${{n}^{mn}}$
4. None of these

Answer 1

Here in this question we have been asked to find the number of ways in which $mn$ letters can be posted in $n$ letter boxes. This question is related to arrangements. Here one letter can be posted in $n$ letter boxes.

Complete step-by-step solution:
Now considering from the question we have been asked to find the number of ways in which $mn$ letters can be posted in $n$ letter boxes.
By clearly observing the given data, here we can say that one letter can be posted in $n$ letter boxes.
Similarly each and every letter in the given set can be posted in $n$ letter boxes.
Hence we can say that the number of ways in which $mn$ letters can be posted in $n$ letter boxes will be given as $mn\times mn\times mn\times ..............\times mn\left( n\text{ times} \right)$ .
This can be further simplified and written as $\Rightarrow {{\left( mn \right)}^{n}}$ .
This question is related to the concept of arrangements, that is, permutations and combinations.
Therefore we can conclude that the number of ways in which $mn$ letters can be posted in $n$ letter boxes will be given as ${{\left( mn \right)}^{n}}$ .
Hence we will mark the option “1” as correct.

Note: While answering questions of this type we should be sure with our concepts that we are going to apply in between the steps. Many of us generally get confused and mark the option “1” as correct because they consider the number of ways as $n\times n\times n\times ............\times n\left( mn\text{ times} \right)$ which is a wrong answer clearly.