Question

In how many ways can three letters be posted in four letterboxes in a village? If all three letters are not posted in the same letterbox, find the corresponding number of ways of posting.

Answer 1

Hint:In this question, we first need to find the number of ways we can put the first letter in the four boxes, then the second letter and then the third letter and then multiply all of them to get the total number of ways. Now, for all letters to be not pasted in the same letter box we need to subtract the ways in which all the letters are being posted in the same box from the total number of possible ways.

Complete step-by-step answer:
Now, given in the question that there are three letters and four letterboxes
Let us now place the first letter and check the number of ways possible
Now, this letter can be placed in any of the four boxes which gives
$\Rightarrow 4\text{ ways}$
Now, the second letter again to be posted has four boxes available which can be posted in
$\Rightarrow 4\text{ ways}$
Now, again the third letter also has four boxes available which has
$\Rightarrow 4\text{ ways}$
Now, the total number of ways in which these three letters can be posted is given by
$\Rightarrow 4\times 4\times 4$
Now, on further simplification we get,
$\Rightarrow 64$
Thus, three letters can be posted in four letterboxes in 64 ways.
Now, we need to place three letters in four boxes provided that all the letters not to be placed in one box
Here, for all the letters to be placed in the same box we have different possibilities
Now, for all the boxes to be placed in the first box we have 1 possible way
In the same way, for all the letters to be placed in the second box we have 1 way, to be placed in the third box we have 1 way and then in the fourth box we have 1 way.
Thus, for all the letters to be placed in the same box we have 4 ways.
Now, on subtracting these 4 ways of all the letters being placed in 1 box from the total number of ways we get the value for them to be not placed in 1 box
$\Rightarrow 64-4$
$\Rightarrow 60$
Thus, the number of ways for all three letters not posted in the same letterbox are 60.

Note:Instead of subtracting the ways for all letters to be placed in one box from total number of ways we can also solve it by finding the ways in which all the letters can be placed in different boxes and two letters in one box and the remaining one in the other and then all add these possibilities to get the result.It is important to note that while finding the total number of ways we need to consider 4 possible ways for all the three letters as there is no condition given in particular that there can't be more than one in one box.